Expected values of topological descriptors for possible kink chains of type 2⊤2

Chen, Ruxian; Razzaque, Asima; Khalil, Maham; Kanwal, Salma; Noor, Saima; Nazir, Robina

doi:10.3389/fchem.2024.1517892

ORIGINAL RESEARCH article

Front. Chem. , 18 February 2025

Sec. Theoretical and Computational Chemistry

Volume 12 - 2024 | https://doi.org/10.3389/fchem.2024.1517892

Expected values of topological descriptors for possible kink chains of type ₂⊤2

Ruxian Chen^1,2

Asima Razzaque^3,4*

Maham Khalil⁵

Salma Kanwal⁵*

Saima Noor^3,4

Robina Nazir^3,4

¹School of Public and General Education, Guangzhou Civil Aviation College, Guangzhou, Gaudong, China
²Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
³Department of Basic Science, Preparatory Year, King Faisal University, Al-Ahsa, Saudi Arabia
⁴Department of Mathematics, College of Science, King Faisal University, Al-Ahsa, Saudi Arabia
⁵Department of Mathematics, Lahore College for Women University, Lahore, Pakistan

In this paper, we investigate square-hexagonal chains, a class of systems where the inner dual of a structure with a square-hexagon shape forms a path graph. The specific configuration of square and hexagonal polygons, and how they are concatenated, leads to different types of square-hexagonal chains. A square containing a vertex of degree 2 is classified as having a kink, and the resulting kink is referred to as a type ${}_{2}⊤$ kink. This kink is further subdivided into three types: ${}_{2}⊤_{1}$ , ₂ $⊤ 2$ , and ₂ $⊤ 3$ . We focus on the kink chain of type ₂ $⊤ 2$ and compute various topological descriptors for this configuration. By deriving analytical expressions, we determine the maximizing and minimizing values of these descriptors. Additionally, we provide a comprehensive analysis of the expected values for these descriptors and offer a comparison of their behaviors through analytical, numerical, and graphical methods. These results offer insights into the structural properties and behavior of square-hexagonal chains, particularly in relation to the optimization of topological descriptors.

1 Preliminaries

Graph theory is a mathematical discipline that studies graphs, which are abstract structures used to model and analyze relationships between objects. A Graph $ζ = (V (ζ), E (ζ))$ is defined to be the collection of vertices (nodes) and edges (links or arcs) where $V (ζ)$ and $E (ζ)$ denotes set of vertices and edges respectively. The order of a graph, denoted as $n = | V (ζ)) |$ refers to the total number of vertices in the graph. The size of a graph, denoted as $m = | E (ζ) |$ refers to the total number of edges in the graph. The degree of a vertex, denoted by $d e g (w)$ or $d_{w}$ , is the number of edges connected to that vertex. The distance $d (w, x)$ between two vertices $w, x$ is the length of the shortest path joining them. For basic definitions related to graph theory, we refer (Trinajstic, 1992).

In chemical graph theory, the numerical values assigned to a molecular graph, known as topological indices or molecular descriptors, are often used to correlate with chemical structures and their properties. In other words, topological indices refer to graph invariants or descriptors that have significant chemical relevance. These indices are based on the graphical representation of a molecule and can encode chemical information such as atom types and bond multiplicities. Topological indices are valuable for predicting specific chemical and physical properties of the underlying molecular structure, combining logical and mathematical principles to translate a molecule’s symbolic representation into a useable numerical form. Chemical graph theory, which merges the fields of chemistry and graph theory, uses graphs to represent chemical structures, providing insights into the physical and chemical characteristics of molecules.

The first degree based topological descriptor was introduced by Milan Randic in 1975 in his paper (Randic, 1975) “On characterization of molecular branching.” This index is referred to as Randic index and is defined as

R (ζ) = \sum_{w x \in E (ζ)} \frac{1}{\sqrt{d_{w} + d_{x}}} .

The Randic index has been recognized as a valuable tool in drug design and has been widely used for this purpose in various studies (Randic, 1975).

The first and second Zagreb indices are the oldest degree based graph invariants introduced by Gutman and Trinajstic (Gutman and Trinajstic, 1972b) in 1972. They were later included among topological descriptors and are defined as

M_{1} (ζ) = \sum_{w x \in E (ζ)} (d_{w} + d_{x}) .

M_{2} (ζ) = \sum_{w x \in E (ζ)} (d_{w} \times d_{x}) .

The first and second Zagreb indices were initially applied to branching problem (Gutman et al., 1975b). Later, they found applications in QSPR and QSAR studies (Balaban, 1979; Bonchev and Trinajsti, 2001; Devillers and Balaban, 1999).

The applicability of Zagreb indices motivated the researchers to define different variants of Zagreb indices. The Hyper Zagreb index was put forwarded by Shirdel et al. (Shirdel et al., 2013) in 2013 and is defined as

H Z (ζ) = \sum_{w x \in E (ζ)} {(d_{w} + d_{x})}^{2} .

Another variant of Zagreb indices namely, first and second redefined Zagreb indices were introduced by Ranjini et al. (Ranjini et al., 2013)

R e Z_{1} (ζ) = \sum_{w x \in E (ζ)} (\frac{d_{w} + d_{x}}{d_{w} \cdot d_{x}}) .

R e Z_{2} (ζ) = \sum_{w x \in E (ζ)} (\frac{d_{w} \cdot d_{x}}{d_{w} + d_{x}}) .

Motivated by the definitions of first and second Zagreb indices and their chemical applicability, V. Kulli (Kulli, 2017a) introduces the first and second Gourava indices. These topological indices are defined as

G O_{1} (ζ) = \sum_{w x \in E (ζ)} (d_{w} + d_{x} + d_{w} \cdot d_{x}) .

G O_{2} (ζ) = \sum_{w x \in E (ζ)} (d_{w} + d_{x}) (d_{w} \cdot d_{x}) .

The first and second Revan descriptors were introduced by V. Kulli (Kulli, 2017b) and are defined as

R_{1} (ζ) = \sum_{w x \in E (ζ)} r_{w} + r_{x}

R_{2} (ζ) = \sum_{w x \in E (ζ)} r_{w} \cdot r_{x}

where $r_{w}$ is defined as $r_{w} = Δ_{ζ} + δ_{ζ} - d_{w}$ , where $Δ$ and $δ$ denotes the maximum and minimum degree among the vertices of $ζ$ .

For more details on the importance of topological indices and their applications see (Noreen and Mahmood, 2018; Wei and Shiu, 2019; Raza, 2020; Wei et al., 2020; Fang et al., 2021; Alraqad et al., 2022; Zhang X. et al., 2023; Zhang Guoping et al., 2023; Hui et al., 2023a; Hui et al., 2023b; Huang et al., 2023). For results related to mathematical properties of the topological indices, we refere (Zhou, 2004; Zhao et al., 2016; Gao et al., 2017; Kulli, 2017c; Kulli, 2017e; Zhang et al., 2024; Govardhan et al., 2024; Prabhu et al., 2024a; Prabhu et al., 2024b).

2 Square-hexagonal system and kink chains

A square-hexagonal system, also known as a rectangular hexagonal system, is a connected geometric structure created by joining equal-sized squares and hexagons. This arrangement blends elements of square and hexagonal lattices, forming a distinctive repeating pattern that combines the characteristics of both shapes. The lattice points in this system create a regular and continuous design, where each polygon is linked to its neighbors. Two polygons are considered neighboring if they share a common edge, emphasizing the interconnected nature of this hybrid structure. This system is widely used in crystallography and materials science, particularly for analyzing the structures of materials with hexagonal crystal systems that exhibit square symmetry along specific crystallographic directions. It provides a geometric framework for understanding the arrangement of atoms, ions, or other structural components within such materials.

A square-hexagonal system is a two-dimensional lattice structure that combines square and hexagonal elements in a unified arrangement. In contrast, a square-hexagonal chain is a one-dimensional sequence where square and hexagonal configurations alternate along its length. While both concepts incorporate square and hexagonal features, they differ in their structure and intended applications. The structure of a square-hexagonal chain varies depending on the types of polygons used and how they are concatenated. A square-hexagonal chain composed of $n$ polygons is denoted as $R_{n}$ . If all the polygons in $R_{n}$ are squares, it is referred to as a polyomino chain (Li et al., 2023). Similarly, if all the polygons are hexagons, $R_{n}$ is called a hexagonal chain (Alraqad et al., 2022). However, when squares and hexagons alternate in the chain, $R_{n}$ is specifically known as a phenylene chain. (as in (Raza, 2021; Shooshtari et al., 2022).

To derive key results, it is important to introduce certain terminologies related to square-hexagonal chains. In graph theory, a kink refers to a point in the graph where there is a sudden change in direction or slope. More precisely, a kink is a vertex whose degree is greater than the degrees of its neighboring vertices, resulting in a bend or angular deviation in the graph’s structure.

Kinks play a significant role in graph analysis, as they often highlight structural changes or key points within the graph. They can influence various graph properties and algorithms, including traversal methods, connectivity analysis, and the identification of critical nodes or hubs in networks. In network analysis, for instance, identifying kinks or high-degree vertices can reveal essential nodes that contribute significantly to the network’s connectivity or exert considerable influence. Moreover, the presence of kinks can affect processes like random walks, as high-degree vertices are more likely to attract repeated visits, thereby altering the overall dynamics of the system.

A polygon at one end of a chain, typically lacking a neighboring polygon on one of its sides, is referred to as a terminal polygon. In contrast, a polygon located within the chain, with neighboring polygons on both sides and not positioned at the chain’s ends, is termed a non-terminal polygon.

If the centers of two adjacent non-terminal polygons are not collinear, the polygon is described as kinked in the chain. There are two types of square-hexagonal kinks, denoted as ${}_{1}⊤$ and ${}_{2}⊤$ (Alraqad et al., 2022). In type ${}_{1}⊤$ , the kink is formed by a hexagon, while in type ${}_{2}⊤$ , square occurs as a kink. A non-terminal hexagon is considered kinked if and only if it contains two consecutive vertices of degree two. Similarly, a non-terminal square is considered kinked if and only if it has a single vertex with a degree of two. Following (Alraqad et al., 2022), we focus on square-hexagonal chains related to the kinks described below:

(1) ${K i n k s}_{1} ⊤$ ; A non-terminal hexagon that has exactly two vertices with a degree of two.

(2) ${K i n k s}_{2} ⊤_{1}$ : A non-terminal square that is adjacent to two squares and has a vertex with a degree of two. (Figure 1A);

(3) ${K i n k s}_{2} ⊤_{2}$ : A non-terminal square that is adjacent to a square and a hexagon and has a vertex of degree two. (Figure 1B);

(4) ${K i n k s}_{2} ⊤_{3}$ : A non-terminal square adjacent to two hexagons and has a vertex of degree two. (Figure 1C);

Figure 1

Figure 1. Kinks of Type ${}_{2}⊤_{1}$ , ${}_{2}⊤_{2}$ and ${}_{2}⊤_{3}$ , (A) kink1, (B) kink2, (C) kink3.

In graph theory, the expected values of topological indices serve as statistical measures of a graph’s structural properties, capturing key characteristics such as connectivity, distances, and vertex degrees. These values are particularly valuable for analyzing and comparing random graphs, optimizing network designs, and predicting behaviors in fields like chemistry, biology, and social networks. Additionally, they enable the development of efficient algorithms for large-scale graph analysis by reducing the computational complexity of calculating indices across various graph models. Our work is motivated from our previous work on the kink chains introduced in (Chunsong et al., 2024). We considered three types of kink chains, ${}_{2}⊤_{2}$ , which are categorized as ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ , and ${}_{2}⊤_{2}^{3}$ , based on the specific way squares and hexagons are concatenated. For generality, we divided our analysis into two cases: odd and even numbered kink chains, addressing their orders, sizes, and corresponding vertex and edge partitions. Additionally, we calculated the topological indices defined earlier and demonstrated that the second Gourava topological index is a maximizing index, while the redefined first Zagreb index is a minimizing index in both cases. Now, we will determine the expected values of topological descriptors for the newly identified kink chains ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ , and ${}_{2}⊤_{2}^{3}$ . These kink chains are defined as;

$•$ Kink chains of. $T y p e {}_{2}⊤_{2}$

(a) $K i n k c h a i n {}_{2}⊤_{2}^{1} :$ A kink chain of type ${}_{2}⊤_{2}$ in which no two adjacent vertices in the hexagons have a degree of 2, except at the terminal polygons. It is represented in Figure 2.

(b) $K i n k c h a i n {}_{2}⊤_{2}^{2} :$ A kink chain of type ${}_{2}⊤_{2}$ in which there are exactly two adjacent vertices with a degree of 2 in the hexagons, excluding the terminal polygons. It is represented in Figure 3.

(c) $K i n k c h a i n {}_{2}⊤_{2}^{3} :$ A kink chain of type ${}_{2}⊤_{2}$ in which there are three adjacent vertices with a degree of 2 in the hexagons, excluding the terminal polygons. It is represented in Figure 4.

Figure 2

Figure 2. $K i n k c h a i n {}_{2}⊤_{2}^{1}$ .

Figure 3

Figure 3. $K i n k c h a i n {}_{2}⊤_{2}^{2}$ .

Figure 4

Figure 4. $K i n k c h a i n {}_{2}⊤_{2}^{3}$ .

Observe that there is no edge between two adjacent vertices of degree 2, only one edge between two adjacent vertices of degree 2, and two edges between two adjacent vertices of degree two in kink chains ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ respectively, except at terminal polygons. Let $\overset{́}{κ}$ represents the number of kinks (squares forming a kink). It is observed that even-numbered kink chains result when two squares are joined at the terminal, while odd-numbered kink chains occur when a hexagon is attached at the terminal. Consequently, an odd-numbered chain corresponds to a square terminal, whereas an even-numbered chain corresponds to a hexagonal terminal. These cases are mathematically represented as $\overset{́}{κ} = 2 n - 1$ for odd-numbered chains and $\overset{́}{κ} = 2 n$ for even-numbered chains, where $n \in N$ . The order $p$ of each kink chain is the same across these types and follows the formula $p = 6 + 4 \overset{́}{κ}$ . However, the size $q$ of the chain differs: for $\overset{́}{κ} = 2 n - 1$ (terminal square), $q = \frac{11 \overset{́}{κ} + 13}{2}$ , and for $\overset{́}{κ} = 2 n$ (terminal hexagon), $q = \frac{11 \overset{́}{κ} + 12}{2}$ .

2.1 Vertex and corresponding edge partitions of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

Let $E_{i j} = {e = w x; d_{w} = i, d_{x} = j}$ be the subclass of edge sets of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ then $| E_{i j} |$ depends on number of kinks $\overset{́}{κ}$ . Note that there are only $(2,2)$ , $(2,3)$ , $(2,4)$ , $(3,4)$ and $(4,4)$ -type of edges in each kink chain. Table 1 represents the edge partitions of each kink chain accordingly.

Table 1

Table 1. Edge partitions of ${}_{2}{⊤_{2}^{1}}$ , ${}_{2}{⊤_{2}^{2}}$ and ${}_{2}{⊤_{2}^{3}}$ ; $n \in N$ .

Let $V_{i} = {w \in V (G) | d_{w} = i}$ be the subclass of vertex sets of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ . $| V_{i} |$ depends on number of kinks $\overset{́}{κ}$ . There are only vertices of degree 2, 3 and 4 in each kink chain. Table 2 represents vertex partitions of chains in both cases. Note that the vertex partitions remain same for each kink chain.

Table 2

Table 2. Vertex partitions of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ .

3 Topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

In this section, we will calculate some topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ using Tables 1, 2. Let $p$ denotes the kink chain for kinks of type ${}_{2}⊤_{2}$ , where p varies from 1 to 3.

Lemma 1. For $n \in N$ the first Revan index of $Kink chain {}_{2}⊤_{2}^{p}$ is given as;

R_{1} ({}_{2}⊤_{2}^{p}) = \{\begin{cases} 33 \overset{́}{κ} + 49 if p = 1,2; f o r \overset{́}{κ} = 2 n - 1 \end{cases}

R_{1} ({}_{2}⊤_{2}^{p}) = \{\begin{cases} 33 \overset{́}{κ} + 48 if p = 1,2; f o r \overset{́}{κ} = 2 n \end{cases}

R_{1} ({}_{2}⊤_{2}^{p}) = \{\begin{cases} 32 \overset{́}{κ} + 50 if p = 3 \end{cases}

Proof. For $\overset{́}{κ} = 2 n - 1$

$• p = 1$

\begin{aligned} R_{1} ({}_{2}⊤_{2}^{1}) = 8 (4) + 7 (2 \overset{́}{κ} + 2) + 6 (2 \overset{́}{κ}) + 5 (\overset{́}{κ} + 1) + 4 (\frac{\overset{́}{κ} - 1}{2}) = 33 \overset{́}{κ} + 49 \end{aligned}

$• p = 2$

\begin{matrix} R_{1} ({}_{2}⊤_{2}^{2}) = 8 (\frac{\overset{́}{κ} + 7}{2}) + 7 (\frac{3 \overset{́}{κ} + 5}{2}) + 6 (\frac{3 \overset{́}{κ} + 1}{2}) + 5 (\frac{3 \overset{́}{κ} + 1}{2}) + 4 (\frac{\overset{́}{κ} - 1}{2}) = 33 \overset{́}{κ} + 49 \end{matrix}

$• p = 3$

\begin{aligned} R_{1} ({}_{2}⊤_{2}^{3}) = 8 (\overset{́}{κ} + 3) + 7 (4) + 6 (3 \overset{́}{κ} - 1) + 5 (2) + 4 (\frac{3 \overset{́}{κ} - 3}{2}) = 32 \overset{́}{κ} + 50 \end{aligned}

For $\overset{́}{κ} = 2 n$

$• p = 1$

\begin{matrix} R_{1} ({}_{2}⊤_{2}^{1}) = 8 (6) + 7 (2 \overset{́}{κ}) + 6 (2 \overset{́}{κ}) + 5 (\overset{́}{κ}) + 4 (\frac{\overset{́}{κ}}{2}) = 33 \overset{́}{κ} + 48 \end{matrix}

$• p = 2$

\begin{aligned} R_{1} ({}_{2}⊤_{2}^{2}) = 8 (\frac{\overset{́}{κ} + 10}{2}) + 7 (\frac{3 \overset{́}{κ} + 2}{2}) + 6 (\frac{3 \overset{́}{κ} + 2}{2}) + 5 (\frac{3 \overset{́}{κ} - 2}{2}) + 4 (\frac{\overset{́}{κ}}{2}) = 33 \overset{́}{κ} + 48 \end{aligned}

$• p = 3$

\begin{aligned} R_{1} ({}_{2}⊤_{2}^{3}) = 8 (\overset{́}{κ} + 4) + 7 (4) + 6 (3 \overset{́}{κ} - 2) + 5 (2) + 4 (\frac{3 \overset{́}{κ} - 4}{2}) = 32 \overset{́}{κ} + 50 \end{aligned}

Lemma 2. For $n \in N$ the $2^{n d}$ Revan, $1^{s t}$ and $2^{n d}$ Redefined Zagreb, Hyper-Zagreb, $1^{s t}$ and $2^{n d}$ Gourava descriptors of kink chains ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ are are presented in Table 3;

Table 3

Table 3. Topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ ; $n \in N$ .

4 Graphical representation of numerical values of topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

In this section, we compared the above calculated topological descriptors using graphical representation of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ at different values of $\overset{́}{κ}$ for odd and even numbered kink chains. From the Figures 5–11 we conclude that $G O_{2}$ descriptor of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ hits a highest value for both $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ . It follows that $G O_{2}$ is a maximizing descriptor. On the other hand $R e Z_{1}$ descriptor of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ reaches a lowest value for both $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ , thus $R e Z_{1}$ is a minimizing descriptor.

Figure 5

Figure 5. Graphical representation of $R_{1} ({}_{2}⊤_{2}^{1})$ , $R_{1} ({}_{2}⊤_{2}^{2})$ and $R_{1} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ , (B) For $\overset{́}{κ} = 2 n$ .

Figure 6

Figure 6. Graphical representation of $R_{2} ({}_{2}⊤_{2}^{1})$ , $R_{2} ({}_{2}⊤_{2}^{2})$ and $R_{2} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$ .

Figure 7

Figure 7. Graphical representation of $R e Z_{1} ({}_{2}⊤_{2}^{1})$ , $R e Z_{1} ({}_{2}⊤_{2}^{2})$ and $R e Z_{1} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$ .

Figure 8

Figure 8. Graphical representation of $R e Z_{2} ({}_{2}⊤_{2}^{1})$ , $R e Z_{2} ({}_{2}⊤_{2}^{2})$ and $R e Z_{2} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$

Figure 9

Figure 9. Graphical representation of $H Z ({}_{2}⊤_{2}^{1})$ , $H Z ({}_{2}⊤_{2}^{2})$ and $H Z ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$ .

Figure 10

Figure 10. Graphical representation of $G O_{1} ({}_{2}⊤_{2}^{1})$ , $G O_{1} ({}_{2}⊤_{2}^{2})$ and $G O_{1} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$ .

Figure 11

Figure 11. Graphical representation of $G O_{2} ({}_{2}⊤_{2}^{1})$ , $G O_{2} ({}_{2}⊤_{2}^{2})$ and $G O_{2} ({}_{2}⊤_{2}^{3})$ . (A) For $\overset{́}{κ} = 2 n - 1$ . (B) For $\overset{́}{κ} = 2 n$ .

5 Expected values of topological descriptors of kink chains of type ${}_{2}⊤_{2}$

As we know that there are only three possible kink chains ( ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ ) or arrangements of type ${}_{2}⊤_{2}$ , holding the conditions to make kink at each step. The kink chains for $\overset{́}{κ} = 1$ and $\overset{́}{κ} = 2$ are shown in Figure 12. For $\overset{́}{κ} \geq 2$ , terminal polygons are attachable in three different ways, resulting in three types ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ . Considering that $γ$ represents the probability of attaching terminal polygons in the first or second kind of arrangement, $1 - 2 γ$ represents the probability of attaching the terminal polygon in the third type of arrangement. Possible arrangements of kink chains of type ${}_{2}⊤_{2}$ are shown in Figure 12.

Figure 12

Figure 12. Possible arrangements of kink chains of type ${}_{2}⊤_{2}$ .

Let the $K i n k c h a i n {}_{2}⊤_{2}$ with $\overset{́}{κ}$ number of kinks and probability $γ$ is represented by ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ . We compute Revan, Hyper-Zagreb, redefined Zagreb and Guorava descriptors of possible square-hexagonal kink chains ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ . Let $| E_{i j} | = β_{i j}$ denotes the number of edges for ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ with end vertices of degree i and j respectively. There are only $β_{22}$ , $β_{23}$ , $β_{24}$ , $β_{34}$ and $β_{44}$ -type of edges in ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ . Here $Δ_{{}_{2}⊤_{2}}$ and $δ_{{}_{2}⊤_{2}}$ are 4 and 2 implies that (Revan degrees) $r_{s} = 4,3,2$ for $d_{s} = 2,3$ and 4 respecively. From the definition, the topological descriptors can be expressed as

R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = 8 β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 7 β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 6 β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 5 β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 4 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 12 β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 8 β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 6 β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 4 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{5}{6} β_{23} ({({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{3}{4} β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{7}{12} β_{34} ({({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{1}{2} β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{6}{5} β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{4}{3} β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + \frac{12}{7} β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 2 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 25 β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 36 β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 49 β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 64 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = 8 β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 11 β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 14 β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 19 β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 24 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 30 β_{23} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 48 β_{24} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 84 β_{34} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}} + 128 β_{44} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}

As ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ is a random kink chain, it follows $R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ , $R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ , $R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ , $R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ , $H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ , $G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ and $G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ are random variables. Let us denote by $E_{n}^{R_{1}} = E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ , $E_{n}^{R_{2}} = E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ , $E_{n}^{R e Z_{1}} = E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ , $E_{n}^{R e Z_{2}} = E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ , $E_{n}^{H Z} = E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ , $E_{n}^{G O_{1}} = E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ and $E_{n}^{G O_{2}} = E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]$ the expected values of these descriptors respectively.

Note that if $\overset{́}{κ}$ is odd in kink chains ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ , then at ${(\overset{́}{κ} - 1)}^{t h}$ step, even numbered kink chains are formed, and at ${(\overset{́}{κ} - 2)}^{t h}$ step, odd numbered kink chains are formed. Similarly, if $\overset{́}{κ}$ is even in kink chains ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ at ${(\overset{́}{κ} - 1)}^{t h}$ step, odd numbered kink chains are obtained and at ${(\overset{́}{κ} - 2)}^{t h}$ step, even numbered kink chains are obtained again. So, we furthur divide our results in two possible stages;

(1) At ${(\overset{́}{κ} - 1)}^{t h}$ stage and (2) At ${(\overset{́}{κ} - 2)}^{t h}$ stage.

5.1 Results at ${(\overset{́}{κ} - 1)}^{t h}$ stage

The three possible constructions at ${(\overset{́}{κ} - 1)}^{t h}$ step are as follows:

$1 . {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$

$2 . {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$

$3 . {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$

For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $n \in N$ the change in edge partitions of ${({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}$ for three possible constructions at ${(\overset{́}{κ} - 1)}^{t h}$ step are shown in Tables 4, 5 respectively.

Table 4

Table 4. Change in edge partitions of ${({}_{2}⊤_{2}^{p})}_{\overset{́}{κ}}$ at ${(k - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n - 1$ ; $n \in N$ .

Table 5

Table 5. Change in edge partitions of ${({}_{2}⊤_{2}^{p})}_{\overset{́}{κ}}$ at ${(k - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n$ ; $n \in N$ .

Theorem 5.1. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks.

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{n}^{R_{1}} = \overset{́}{κ} [4 γ + 32] - 4 γ + 50

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{n}^{R_{1}} = 32 \overset{́}{κ} + 50

Proof. $•$ (a). When terminal polygon is a square.

For $\overset{́}{κ} = 1 \Rightarrow n = 1$ , $E_{1} = 82$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , then using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{aligned} R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2] + 7 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 5 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 4 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{aligned}

R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 34 (5.1)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{aligned} R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1] + 7 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + 5 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 4 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{aligned}

R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 34 (5.2)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 8 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 7 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 5 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32 (5.3)

Thus, we have

E_{\overset{́}{κ}}^{R_{1}} = γ R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.1), (6.2) and (6.3), we get the following relation

E_{\overset{́}{κ}}^{R_{1}} = γ [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 34] + γ [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 34] + (1 - 2 γ) [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32]

E_{\overset{́}{κ}}^{R_{1}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 γ + 32

Applying operator E on both sides, we get

E (E_{\overset{́}{κ}}^{R_{1}}) = E_{\overset{́}{κ}}^{R_{1}}

E_{\overset{́}{κ}}^{R_{1}} = E_{\overset{́}{κ} - 1}^{R_{1}} + 4 γ + 32

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{aligned} E_{\overset{́}{κ}}^{R_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{R_{1}} + (\overset{́}{κ} - 1) [4 γ + 32] = 82 + (\overset{́}{κ} - 1) [4 γ + 32] \\ = \overset{́}{κ} [4 γ + 32] - 4 γ + 50 \end{aligned}

$•$ (b). When terminal polygon is a hexagon.

For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 114$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , then using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 7 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 5 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32 (5.4)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 7 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 5 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32 (5.5)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 6 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 7 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 6 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 5 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32 (5.6)

Thus, we have

E_{\overset{́}{κ}}^{R_{1}} = γ R_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.4), (6.5) and (6.6), we get the following relation

E_{\overset{́}{κ}}^{R_{1}} = γ [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32] + γ [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32] + (1 - 2 γ) [R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32]

E_{\overset{́}{κ}}^{R_{1}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 32

Applying operator E on both sides, we get

E (E_{\overset{́}{κ}}^{R_{1}}) = E_{\overset{́}{κ}}^{R_{1}}

E_{\overset{́}{κ}}^{R_{1}} = E_{\overset{́}{κ} - 1}^{R_{1}} + 32

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{aligned} E_{\overset{́}{κ}}^{R_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{R_{1}} + (\overset{́}{κ} - 2) 32 = 114 + (\overset{́}{κ} - 2) 32 \\ = 32 \overset{́}{κ} + 50 \end{aligned}

which completes the proof.

Theorem 5.2. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks. Then

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{\overset{́}{κ}}^{R_{2}} = \overset{́}{κ} [10 γ + 40] - 10 γ + 100

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{\overset{́}{κ}}^{R_{2}} = 52 \overset{́}{κ} + 88

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = 140$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{n - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{aligned} R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2] + 12 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 6 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 4 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{aligned}

R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 44 (5.7)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{aligned} R_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1] + 12 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + 6 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 4 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{aligned}

R_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 46 (5.8)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 12 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 6 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 40 (5.9)

Thus, we have

E_{\overset{́}{κ}}^{R_{2}} = γ R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.7), (6.8) and (6.9), we get the following relation

E_{\overset{́}{κ}}^{R_{2}} = γ [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 44] + γ [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 46] + (1 - 2 γ) [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 40]

E_{\overset{́}{κ}}^{R_{2}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 10 γ + 40

E_{\overset{́}{κ}}^{R_{2}} = E_{\overset{́}{κ} - 1}^{R_{2}} + 10 γ + 40

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{aligned} E_{\overset{́}{κ}}^{R_{2}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{R_{2}} + (\overset{́}{κ} - 1) [10 γ + 40] = 140 + (\overset{́}{κ} - 1) [10 γ + 40] \\ = \overset{́}{κ} [10 γ + 40] - 10 γ + 100 \end{aligned}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 192$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 12 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 6 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52 (5.10)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

R_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 12 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 6 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52 (5.11)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 12 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 8 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 6 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52 (5.12)

Thus, we have

E_{\overset{́}{κ}}^{R_{2}} = γ R_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R_{2} {({}^{2}+^{2} {}_{2}⊤_{2})}_{\overset{́}{κ}} (1 - 2 γ) R_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.10), (6.11) and (6.12), we get the following relation

E_{\overset{́}{κ}}^{R_{2}} = γ [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52] + γ [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52] + (1 - 2 γ) [R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52]

E_{\overset{́}{κ}}^{R_{2}} = R_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 52

E_{\overset{́}{κ}}^{R_{2}} = E_{\overset{́}{κ} - 1}^{R_{2}} + 52

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{aligned} E_{\overset{́}{κ}}^{R_{2}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{R_{2}} + (\overset{́}{κ} - 2) 52 = 192 + (\overset{́}{κ} - 2) 52 \\ = 52 \overset{́}{κ} + 88 \end{aligned}

which completes the proof.

Theorem 5.3. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks, then

E_{\overset{́}{κ}}^{R e Z_{1}} = 4 \overset{́}{κ} + 6

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = 10$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2 + \frac{5}{6} [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + \frac{7}{12} [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + \frac{1}{2} β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.13)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1 + \frac{5}{6} [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + \frac{7}{12} [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + \frac{1}{2} β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.14)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{5}{6} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{7}{12} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{1}{2} [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.15)

Thus, we have

E_{\overset{́}{κ}}^{R e Z_{1}} = γ R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.13), (6.14) and (6.15), we get the following relation

E_{\overset{́}{κ}}^{R e Z_{1}} = γ [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + γ [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + (1 - 2 γ) [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4]

E_{\overset{́}{κ}}^{R e Z_{1}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4

Applying operator E on both sides and $∴ E (E_{\overset{́}{κ}}^{R e Z_{1}}) = E_{\overset{́}{κ}}^{R e Z_{1}}$

E_{\overset{́}{κ}}^{R e Z_{1}} = E_{\overset{́}{κ} - 1}^{R e Z_{1}} + 4

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{matrix} E_{\overset{́}{κ}}^{R e Z_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{R e Z 1} + (\overset{́}{κ} - 1) 4 = 10 + (\overset{́}{κ} - 1) 4 \\ = 4 \overset{́}{κ} + 6 \end{matrix}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 14$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2 + \frac{5}{6} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + \frac{7}{12} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \\ + \frac{1}{2} [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] \end{matrix}

R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.16)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{5}{6} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{7}{12} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{1}{2} [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.17)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{5}{6} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{3}{4} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{7}{12} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{1}{2} [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4 (5.18)

Thus, we have

E_{\overset{́}{κ}}^{R e Z_{1}} = γ R e Z_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R e Z_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R e Z_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.16), (6.17) and (6.18), we get the following relation

E_{\overset{́}{κ}}^{R e Z_{1}} = γ [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + γ [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + (1 - 2 γ) [R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4]

E_{\overset{́}{κ}}^{R e Z_{1}} = R e Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4

E_{\overset{́}{κ}}^{R e Z_{1}} = E_{\overset{́}{κ} - 1}^{R e Z_{1}} + 4

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{matrix} E_{\overset{́}{κ}}^{R e Z_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{R e Z_{1}} + (\overset{́}{κ} - 2) (4) = 14 + (\overset{́}{κ} - 2) 4 \\ = 4 \overset{́}{κ} + 6 \end{matrix}

which completes the proof.

Theorem 5.4. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks.

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{\overset{́}{κ}}^{R e Z_{2}} = \overset{́}{κ} [\frac{28}{3} - \frac{73}{105} γ] + \frac{73}{105} γ + \frac{584}{105}

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{\overset{́}{κ}}^{R e Z_{2}} = \frac{20}{3} \overset{́}{κ} + \frac{288}{35}

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = \frac{1564}{105}$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2 + \frac{6}{5} [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + \frac{12}{7} [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 2 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{934}{105} (5.19)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1 + \frac{6}{5} [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + \frac{12}{7} [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 2 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{953}{105} (5.20)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{6}{5} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{12}{7} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{28}{3} (5.21)

Thus, we have

E_{\overset{́}{κ}}^{R e Z_{2}} = γ R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.19), (6.20) and (6.21), we get the following relation

E_{\overset{́}{κ}}^{R e Z_{2}} = γ [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{934}{105}] + γ [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{953}{105}] + (1 - 2 γ) [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{28}{3}]

E_{\overset{́}{κ}}^{R e Z_{2}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - \frac{73}{105} γ + \frac{28}{3}

E_{\overset{́}{κ}}^{R e Z_{2}} = E_{\overset{́}{κ} - 1}^{R e Z_{2}} - \frac{73}{105} γ + \frac{28}{3}

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{matrix} E_{\overset{́}{κ}}^{R e Z_{2}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{R e Z_{2}} + (\overset{́}{κ} - 1) [\frac{28}{3} - \frac{73}{105} γ] = \frac{1564}{105} + (\overset{́}{κ} - 1) [\frac{28}{3} - \frac{73}{105} γ] \\ = \overset{́}{κ} [\frac{28}{3} - \frac{73}{105} γ] + \frac{73}{105} γ + \frac{584}{105} \end{matrix}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = \frac{2264}{105}$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2 + \frac{6}{5} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + \frac{12}{7} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \\ + 2 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] \end{matrix}

R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3} (5.22)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{6}{5} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{12}{7} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3} (5.23)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{6}{5} β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{4}{3} [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + \frac{12}{7} β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3} (5.24)

Thus, we have

E_{\overset{́}{κ}}^{R e Z_{2}} = γ R e Z_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ R e Z_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) R e Z_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.22), (6.23) and (6.24), we get the following relation

E_{\overset{́}{κ}}^{R e Z_{2}} = γ [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3}] + γ [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3}] + (1 - 2 γ) [R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3}]

E_{\overset{́}{κ}}^{R e Z_{2}} = R e Z_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + \frac{20}{3}

Applying operator E on both sides and $∴ E (E_{\overset{́}{κ}}^{R e Z_{2}}) = E_{\overset{́}{κ}}^{R e Z_{2}}$

E_{\overset{́}{κ}}^{R e Z_{2}} = E_{\overset{́}{κ} - 1}^{R e Z_{2}} + \frac{20}{3}

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{matrix} E_{\overset{́}{κ}}^{R e Z_{2}} = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{R e Z_{2}} + (\overset{́}{κ} - 2) \frac{20}{3} = \frac{2264}{105} + (\overset{́}{κ} - 2) \frac{20}{3} \\ = \frac{20}{3} \overset{́}{κ} + \frac{288}{35} \end{matrix}

which completes the proof.

Theorem 5.5. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks.

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{\overset{́}{κ}}^{H Z} = \overset{́}{κ} [272 - 64 γ] + 64 γ + 62

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{\overset{́}{κ}}^{H Z} = 168 \overset{́}{κ} + 166

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = 334$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2] + 25 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 49 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 64 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 238 (5.25)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{matrix} H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1] + 25 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + 49 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 64 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 242 (5.26)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

\begin{matrix} H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} & = 16 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 25 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 49 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \\ + 64 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \end{matrix}

H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 272 (5.27)

Thus, we have

E_{\overset{́}{κ}}^{H Z} = γ H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.25), (6.26) and (6.27), we get the following relation

E_{\overset{́}{κ}}^{H Z} = γ [H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 238] + γ [H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 242] + (1 - 2 γ) [H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 272]

E_{\overset{́}{κ}}^{H Z} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 64 γ + 272

Applying operator E on both sides and $∴ E (E_{\overset{́}{κ}}^{H Z}) = E_{\overset{́}{κ}}^{H Z}$

E_{\overset{́}{κ}}^{H Z} = E_{\overset{́}{κ} - 1}^{H Z} - 64 γ + 272

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{matrix} E_{\overset{́}{κ}}^{H Z} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{H Z} + (\overset{́}{κ} - 1) (272 - 64 γ) = 334 + (\overset{́}{κ} - 1) (272 - 64 γ) \\ = \overset{́}{κ} [272 - 64 γ] + 64 γ + 62 \end{matrix}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 502$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 25 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 49 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \\ + 64 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] \end{matrix}

H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168 (5.28)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 25 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 49 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 64 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168 (5.29)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 25 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 36 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 49 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 64 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168 (5.30)

Thus, we have

E_{\overset{́}{κ}}^{H Z} = γ H Z {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ H Z {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) H Z {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.28), (6.29) and (6.30), we get the following relation

E_{\overset{́}{κ}}^{H Z} = γ [H Z_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168] + γ [H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168] + (1 - 2 γ) [H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168]

E_{\overset{́}{κ}}^{H Z} = H Z {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 168

E_{\overset{́}{κ}}^{H Z} = E_{\overset{́}{κ} - 1}^{H Z} + 168

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{matrix} = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{H Z} + (\overset{́}{κ} - 2) 168 = 502 + (\overset{́}{κ} - 2) 168 \\ = 168 \overset{́}{κ} + 166 \end{matrix}

which completes the proof.

Theorem 5.6. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks.

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{\overset{́}{κ}}^{G O_{1}} = \overset{́}{κ} [104 - 18 γ] + 18 γ + 38

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{\overset{́}{κ}}^{G O_{1}} = 68 \overset{́}{κ} + 74

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = 142$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2] + 11 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 19 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 24 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 94 (5.31)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{matrix} G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1] + 11 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + 19 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 24 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 96 (5.32)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 8 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 11 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 19 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 24 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 104 (5.33)

Thus, we have

E_{\overset{́}{κ}}^{G O_{1}} = γ G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.31), (6.32) and (6.33), we get the following relation

E_{\overset{́}{κ}}^{G O_{1}} = γ [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 94] + γ [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 96] + (1 - 2 γ) [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 104]

E_{\overset{́}{κ}}^{G O_{1}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 18 γ + 104

Applying operator E on both sides and $∴ E (E_{\overset{́}{κ}}^{G O_{1}}) = E_{\overset{́}{κ}}^{G O_{1}}$

E_{\overset{́}{κ}}^{G O_{1}} = E_{\overset{́}{κ} - 1}^{G O 1} - 18 γ + 104

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{matrix} E_{\overset{́}{κ}}^{G O_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{G O_{1}} + (\overset{́}{κ} - 1) [- 18 γ + 104] = 142 + (\overset{́}{κ} - 1) [104 - 18 γ] \\ = \overset{́}{κ} [104 - 18 γ] + 18 γ + 38 \end{matrix}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 210$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 11 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 19 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 24 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68 (5.34)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 11 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 19 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 24 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68 (5.35)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 8 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 11 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 14 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 19 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 24 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68 (5.36)

Thus, we have

E_{\overset{́}{κ}}^{G O_{1}} = γ G O_{1} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ G O_{1} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) G O_{1} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.34), (6.35) and (6.36), we get the following relation

E_{\overset{́}{κ}}^{G O_{1}} = γ [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68] + γ [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68] + (1 - 2 γ) [G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68]

E_{\overset{́}{κ}}^{G O_{1}} = G O_{1} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 68

E_{\overset{́}{κ}}^{G O_{1}} = E_{\overset{́}{κ} - 1}^{G O_{1}} + 68

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{matrix} E_{\overset{́}{κ}}^{G O_{1}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{G O_{1}} + (\overset{́}{κ} - 2) (68) = 210 + (\overset{́}{κ} - 2) 68 \\ = 68 \overset{́}{κ} + 74 \end{matrix}

which completes the proof.

Theorem 5.7. Let ${}_{2}⊤_{2}$ be a square-hexagonal kink chain and $\overset{́}{κ}$ be the number of kinks.

(a) For $\overset{́}{κ} = 2 n - 1$ ; $n \in N$

E_{\overset{́}{κ}}^{G O_{2}} = \overset{́}{κ} [448 - 170 γ] + 170 γ

(b) For $\overset{́}{κ} = 2 n$ ; $n \in N$

E_{\overset{́}{κ}}^{G O_{2}} = 256 \overset{́}{κ} + 192

Proof. $•$ (a). For $n = 1 \Rightarrow \overset{́}{κ} = 1$ , $E_{1} = 448$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 4, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

\begin{matrix} G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 2] + 30 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 84 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] \\ + 128 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 352 (5.37)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

\begin{matrix} G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} & = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 1] + 30 [β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1] + 84 [β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 3] \\ + 128 β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \end{matrix}

G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 374 (5.38)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 16 β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 30 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 4] + 84 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 128 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2]

G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 448 (5.39)

Thus, we have

E_{\overset{́}{κ}}^{G O_{2}} = γ G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.37), (6.38) and (6.39), we get the following relation

E_{\overset{́}{κ}}^{G O_{2}} = γ [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 352] + γ [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 374] + (1 - 2 γ) [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 448]

E_{\overset{́}{κ}}^{G O_{2}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} - 170 γ + 448

Applying operator E on both sides and $∴ E (E_{\overset{́}{κ}}^{G O_{2}}) = E_{\overset{́}{κ}}^{G O_{2}}$

E_{\overset{́}{κ}}^{G O_{2}} = E_{\overset{́}{κ} - 1}^{G O_{2}} - 170 γ + 448

Using recursive relation upto $\overset{́}{κ} - 1$ terms

\begin{matrix} E_{\overset{́}{κ}}^{G O_{2}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 1)}^{G O_{2}} + (\overset{́}{κ} - 1) [- 170 γ + 448] = 448 + (\overset{́}{κ} - 1) [448 - 170 γ] \\ = \overset{́}{κ} [448 - 170 γ] + 170 γ \end{matrix}

$•$ (b). For $n = 1 \Rightarrow \overset{́}{κ} = 2$ , $E_{2} = 704$ , which is indeed true. Let $\overset{́}{κ} \geq 2$ , using Table 5, we get

$1 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}}$ , then

G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 30 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 84 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 128 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256 (5.40)

$2 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}}$ , then

G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 30 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 84 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 128 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256 (5.41)

$3 .$ If ${({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} \to {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}$ , then

G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = 16 [β_{22} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 30 β_{23} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 48 [β_{24} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 2] + 84 β_{34} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 128 [β_{44} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 1]

G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256 (5.42)

Thus, we have

E_{\overset{́}{κ}}^{G O_{2}} = γ G O_{2} {({}_{2}⊤_{2}^{1})}_{\overset{́}{κ}} + γ G O_{2} {({}_{2}⊤_{2}^{2})}_{\overset{́}{κ}} + (1 - 2 γ) G O_{2} {({}_{2}⊤_{2}^{3})}_{\overset{́}{κ}}

Using (6.40), (6.41) and (6.42), we get the following relation

E_{\overset{́}{κ}}^{G O_{2}} = γ [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256] + γ [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256] + (1 - 2 γ) [G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256]

E_{\overset{́}{κ}}^{G O_{2}} = G O_{2} {({}_{2}⊤_{2})}_{\overset{́}{κ} - 1} + 256

E_{\overset{́}{κ}}^{G O_{2}} = E_{\overset{́}{κ} - 1}^{G O_{2}} + 256

Using recursive relation upto $\overset{́}{κ} - 2$ terms

\begin{matrix} E_{\overset{́}{κ}}^{G O_{2}} & = E_{\overset{́}{κ} - (\overset{́}{κ} - 2)}^{G O_{2}} + (\overset{́}{κ} - 2) (256) = 704 + (\overset{́}{κ} - 2) 256 \\ = 256 \overset{́}{κ} + 192 \end{matrix}

which completes the proof.

The expected values $E_{R_{1}}^{\overset{́}{κ}}$ , $E_{R e_{2}}^{\overset{́}{κ}}$ , $E_{H Z}^{\overset{́}{κ}}$ , $E_{R e Z_{2}}^{\overset{́}{κ}}$ , $E_{G O_{1}}^{\overset{́}{κ}}$ and $E_{G O_{2}}^{\overset{́}{κ}}$ descriptors for $\overset{́}{κ} = 2 n - 1$ ; $n \in N$ depend on $γ$ , but the $E_{R e Z_{1}}^{\overset{́}{κ}}$ is independent of $γ$ for both cases. As for the sake of generality we have taken expectations in odd and even cases. Therefore, at ${(\overset{́}{κ} - 1)}^{t h}$ stage the sum of expected values for $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ of a certain topological descriptor of three kink chains is equal to the sum of the average value of topological descriptor of three kink chain for $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ with a constant factor. As expectation of constant is zero, so our results are true. The values of $R_{1}$ , $R_{2}$ , $H Z$ , $R e Z_{2}$ , $G O_{1}$ and $G O_{2}$ descriptors can be computed by using $γ = \frac{1}{3}$ in the above proved theorems.

Corollary 1. Let $\overset{́}{κ} = 2 n - 1$ ; $n \in N$ then at $(\overset{́}{κ} - 1)$ stage;

$• R_{1} = \frac{100}{3} (\overset{́}{κ} - 1) + 82$

$• R_{2} = \frac{130}{3} (\overset{́}{κ} - 1) + 140$

$• R e Z_{2} = \frac{2867}{315} (\overset{́}{κ} - 1) + \frac{1564}{105}$

$• H Z = \frac{752}{3} (\overset{́}{κ} - 1) + 334$

$• G O_{1} = 98 (\overset{́}{κ} - 1) + 142$

$• G O_{2} = \frac{1174}{3} (\overset{́}{κ} - 1) + 448$

Remark 1. The value of $R e Z_{1}$ descriptor at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $n \in N$ is equal and independent of $γ$ , $i . e$ ;

$R e Z_{1} ({}_{2}⊤_{2}^{p}) = 4 \overset{́}{κ} + 6$ ; $p = 1$ , 2 and 3

It is observed that the expected values $E_{R_{1}}^{\overset{́}{κ}}$ , $E_{R e Z_{1}}^{\overset{́}{κ}}$ , $E_{R e_{2}}^{\overset{́}{κ}}$ , $E_{H Z}^{\overset{́}{κ}}$ , $E_{R e Z_{2}}^{\overset{́}{κ}}$ , $E_{G O_{1}}^{\overset{́}{κ}}$ and $E_{G O_{2}}^{\overset{́}{κ}}$ for $\overset{́}{κ} = 2 n$ ; $n \in N$ are independent of $γ$ and depends only on $\overset{́}{κ}$ .

Remark 2. Let $\overset{́}{κ} = 2 n$ ,; $n \in N$ then at $(\overset{́}{κ} - 1)$ stage,

$• R_{1} = 32 \overset{́}{κ} + 50$

$• R_{2} = 52 \overset{́}{κ} + 88$

$• R e Z_{2} = \frac{20}{3} \overset{́}{κ} + \frac{288}{35}$

$• H Z = 168 \overset{́}{κ} + 166$

$• G O_{1} = 68 \overset{́}{κ} + 74$

$• G O_{2} = 256 \overset{́}{κ} + 192$

5.2 Analytical expressions at ${(k - 1)}^{t h}$ stage

Now we analytically prove that at ${(\overset{́}{κ} - 1)}^{t h}$ stage, for any value of $γ$ and $\overset{́}{κ}$ , the $2^{n d}$ Gourava descriptor is always greater than the remaining six descriptors, namely, ( $1^{s t}$ and $2^{n d}$ ) Revan descriptors, ( $1^{s t}$ and $2^{n d}$ ), Zagreb descriptors, Hyper-Zagreb descriptor and $1^{s t}$ Gourava descriptor for $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ . All the expressions holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ .

Corollary 2. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $n \in N$ , we have

E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [448 - 170 γ] + 170 γ\} - \{\overset{́}{κ} [272 - 64 γ] + 64 γ + 62\} \\ = 106 γ (1 - \overset{́}{κ}) + 2 (224 \overset{́}{κ} - 31) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{256 \overset{́}{κ} + 192\} - \{168 \overset{́}{κ} + 166\} \\ = 88 \overset{́}{κ} + 26 > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

Corollary 3. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $\overset{́}{κ} \in N$ , we have

E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [272 - 64 γ] + 64 γ + 62\} - \{\overset{́}{κ} [104 - 18 γ] + 18 γ + 38\} \\ = 46 γ (1 - \overset{́}{κ}) + 2 (136 \overset{́}{κ} - 19) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{168 \overset{́}{κ} + 166\} - \{68 \overset{́}{κ} + 74\} \\ = 100 \overset{́}{κ} + 92 > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

Corollary 4. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $\overset{́}{κ} \in N$ , we have

E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [104 - 18 γ] + 18 γ + 38\} - \{\overset{́}{κ} [10 γ + 40] - 10 γ + 100\} \\ = 28 γ (1 - \overset{́}{κ}) + 2 (32 \overset{́}{κ} - 31) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{68 \overset{́}{κ} + 74\} - \{52 \overset{́}{κ} + 88\} \\ = 16 \overset{́}{κ} - 14 > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

Corollary 5. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $\overset{́}{κ} \in N$ , we have

E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [10 γ + 40] - 10 γ + 100\} - \{\overset{́}{κ} [4 γ + 32] - 4 γ + 50\} \\ = 6 γ (\overset{́}{κ} - 1) + 2 (4 \overset{́}{κ} - 25) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{52 \overset{́}{κ} + 88\} - \{32 \overset{́}{κ} + 50\} \\ = 20 \overset{́}{κ} + 38 > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

Corollary 6. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $\overset{́}{κ} \in N$ , we have

E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [4 γ + 32] - 4 γ + 50\} \\ - \{\overset{́}{κ} [\frac{28}{3} - \frac{73}{105} γ] + \frac{584}{105} + \frac{73}{105} γ\} \\ = \frac{493}{105} γ (\overset{́}{κ} - 1) + \frac{2}{3} (34 \overset{́}{κ} - \frac{2333}{35}) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{32 \overset{́}{κ} + 50\} - \{\frac{20}{3} \overset{́}{κ} + \frac{288}{35}\} \\ = \frac{76}{3} \overset{́}{κ} + \frac{1462}{35} > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

Corollary 7. For $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ ; $\overset{́}{κ} \in N$ , we have

E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}]

Proof. $•$ For $\overset{́}{κ} = 2 n - 1$

\begin{matrix} E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\overset{́}{κ} [\frac{28}{3} - \frac{73}{105} γ] + \frac{584}{105} + \frac{73}{105} γ\} - \{4 \overset{́}{κ} + 6\} \\ = \frac{73}{105} γ (1 - \overset{́}{κ}) + \frac{2}{3} (8 \overset{́}{κ} - \frac{23}{35}) > 0 \end{matrix}

which holds for $γ = \frac{1}{3}$ and for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

$•$ For $\overset{́}{κ} = 2 n$

\begin{matrix} E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] - E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & = \{\frac{20}{3} \overset{́}{κ} + \frac{288}{35}\} - \{4 \overset{́}{κ} + 6\} \\ = \frac{8}{3} \overset{́}{κ} + \frac{78}{35} > 0 \end{matrix}

which holds for all $\overset{́}{κ} \in N$ , so we have

\begin{matrix} E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

From the above analytical expressions we get;

Corollary 8.

\begin{matrix} E [G O_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] & > E [H Z {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [G O_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \\ > E [R_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{2} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] > E [R e Z_{1} {({}_{2}⊤_{2}^{γ})}_{\overset{́}{κ}}] \end{matrix}

5.3 A comparison of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage

Table 6, 7 depict the expected values of Revan descriptors, Zagreb descriptors, Hyper-Zagreb descriptor and Gourava descriptors for $γ = \frac{1}{3}$ and $\overset{́}{κ} = 2 n - 1$ and $\overset{́}{κ} = 2 n$ respectively. Observe that the value of expectation of $2^{n d}$ Gourava descriptor is always greater than the remaining six descriptors in both cases.

Table 6

Table 6. Expected values for topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $γ = \frac{1}{3}$ and $\overset{́}{κ} = 2 n - 1$ .

Table 7

Table 7. Expected values for topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $γ = \frac{1}{3}$ and $\overset{́}{κ} = 2 n$ .

5.4 Graphical representation of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage

The Figures 13–16 shows that expectation of $2^{n d}$ Gourava descriptor attains maximum value and of $1^{s t}$ redefined Zagreb descriptor attains minimum value at ${(\overset{́}{κ} - 1)}^{t h}$ stage for both the cases.

Figure 13

Figure 13. Graphical representation of expectations of $R_{1}$ , $R_{2}$ , $R e Z_{1}$ and $R e Z_{2}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n - 1$ .

Figure 14

Figure 14. Graphical representation of expectations of $R_{1}$ , $R_{2}$ , $R e Z_{1}$ and $R e Z_{2}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n$ .

Figure 15

Figure 15. Graphical representation of expectations of $H Z$ , $G O_{1}$ and $G O_{2}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n - 1$ .

Figure 16

Figure 16. Graphical representation of expectations of $H Z$ , $G O_{1}$ and $G O_{2}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage for $\overset{́}{κ} = 2 n$ .

6 Applications

Kinks, which denote abrupt changes in the direction of edges within a graph, hold notable applications across diverse fields. In circuit design, minimizing kinks optimizes wire lengths and enhances efficiency. Network routing benefits from understanding kinks, as they affect data flow and network performance. Transportation planners use kink analysis to streamline traffic, plan intersections, and design efficient road networks. Graph drawing algorithms consider kinks for aesthetically pleasing and comprehensible visual representations. Lastly, in various applications where visual appeal matters, reducing kinks enhances the attractiveness and clarity of graph representations.

Studying the interforce interactions and scattering of (Lima and Almeida, 2023) kink-antikink-like solutions in a two-dimensional dilaton gravity model has practical implications in fields like material science, nonlinear optics, and cosmology. It aids in understanding energy distribution, stability dynamics, and defect interactions, which are crucial for developing advanced technologies and predicting behaviors in complex physical systems. The in situ (Zhu et al., 2023) investigations have revealed the important role for the kinks. During the growth, the creation of kinks determines the growth rate. Besides, when two domains coalesce, the shape of the final flake is affected by kinks.

In computer graphics, square and hexagonal grids are frequently used to create images or simulations. Kinks in these grids can represent corners or junctions in a digital image. Hexagonal kinks are essential in the study of tessellation and pattern generation. The expected value of random graphs plays a crucial role in graph theory as well. It helps analyze and predict various graph properties in probabilistic settings. By studying expected values in graph theory, we gain a balanced understanding of how certain structures behave under randomness, which informs both theory and practical applications. Expected values are used to calculate the probability that a random graph is connected. It can be employed to estimate the likelihood that two randomly generated graphs are isomorphic. This is valuable in assessing the structural similarity between graphs.

7 Conclusion

In this research work, we determined $R_{1}$ , $R_{2}$ , $R e Z_{1}$ , $R e Z_{2}$ , $H Z$ , $G O_{1}$ and $G O_{2}$ descriptors for the graphical structures of kink chains of type ${}_{2}⊤_{2}$ named as ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ . We infered that $G O_{2}$ is a maximizing, while $R e Z_{1}$ is a minimizing descriptor of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ ,for both, odd and even case. Further we determined expected values of topological descriptors of ${({}_{2}⊤_{2}^{γ})}_{k}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage. We analyzed that value of $R e Z_{1}$ descriptor of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$ and $E_{k}^{R e Z_{1}}$ is same and independent of $γ$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage. We made numerical comparison for these expected values at ${(\overset{́}{κ} - 1)}^{t h}$ stage and conclude that expected value, $E_{k}^{G O_{2}}$ is greater while expected value, $E_{k}^{R e Z_{1}}$ is smaller among other expectations $E_{k}^{R_{1}}$ , $E_{k}^{R_{2}}$ , and $E_{k}^{R e Z_{2}}$ , $E_{k}^{H Z}$ , and $E_{k}^{G O_{1}}$ at ${(\overset{́}{κ} - 1)}^{t h}$ stage. Also, we gave exact analytical expressions of this comparison at ${(\overset{́}{κ} - 1)}^{t h}$ stage which agree with numerical values of comparisons. Results at ${(\overset{́}{κ} - 2)}^{t h}$ stage will be computed in the next article.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

RC: Conceptualization, Investigation, Methodology, Validation, Writing–review and editing. AR: Conceptualization, Investigation, Methodology, Validation, Writing–review and editing. MK: Formal Analysis, Investigation, Validation, Writing–original draft. SK: Conceptualization, Methodology, Validation, Writing–original draft. SN: Conceptualization, Formal Analysis, Investigation, Validation, Writing–review and editing. RN: Conceptualization, Formal Analysis, Investigation, Validation, Writing–review and editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by National Natural Science Foundation of China (No. 62407011, 62172116) and the Guangzhou Academician and Expert Workstation (No. 2024-D003) and Deanship of the Science Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU242800).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: random square-hexagonal kink chains, topological descriptors, expected values, Randic index, Zagreb indices

Citation: Chen R, Razzaque A, Khalil M, Kanwal S, Noor S and Nazir R (2025) Expected values of topological descriptors for possible kink chains of type ₂ $⊤_{2}$ . Front. Chem. 12:1517892. doi: 10.3389/fchem.2024.1517892

Received: 27 October 2024; Accepted: 16 December 2024;
Published: 18 February 2025.

Edited by:

Jose Luis Cabellos, Polytechnic University of Tapachula, Mexico

Reviewed by:

María Guadalupe Hernández-Linares, Benemérita Universidad Autónoma de Puebla, Mexico
Xianya Geng, Anhui University of Science and Technology, China

Copyright © 2025 Chen, Razzaque, Khalil, Kanwal, Noor and Nazir. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Asima Razzaque, YXJhenphcXVlQGtmdS5lZHUuc2E=; Salma Kanwal, c2FsbWEua2Fud2FsQGxjd3UuZWR1LnBr

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

Expected values of topological descriptors for possible kink chains of type ₂⊤2

1 Preliminaries

2 Square-hexagonal system and kink chains

2.1 Vertex and corresponding edge partitions of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

3 Topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

4 Graphical representation of numerical values of topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

5 Expected values of topological descriptors of kink chains of type ${}_{2}⊤_{2}$

5.1 Results at ${(\overset{́}{κ} - 1)}^{t h}$ stage

5.2 Analytical expressions at ${(k - 1)}^{t h}$ stage

5.3 A comparison of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage

5.4 Graphical representation of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage

6 Applications

7 Conclusion

Data availability statement

Author contributions

Funding

Conflict of interest

Generative AI statement

Publisher’s note

References

94% of researchers rate our articles as excellent or good

94% of researchers rate our articles as excellent or good

Expected values of topological descriptors for possible kink chains of type 2⊤2

1 Preliminaries

2 Square-hexagonal system and kink chains

2.1 Vertex and corresponding edge partitions of ⊤212,⊤222 and ⊤232

3 Topological descriptors of ⊤212, ⊤222 and ⊤232

4 Graphical representation of numerical values of topological descriptors of ⊤212, ⊤222 and ⊤232

5 Expected values of topological descriptors of kink chains of type ⊤22

5.1 Results at (κ́−1)th stage

5.2 Analytical expressions at (k−1)th stage

5.3 A comparison of expected values of topological descriptors at (κ́−1)th stage

5.4 Graphical representation of expected values of topological descriptors at (κ́−1)th stage

6 Applications

7 Conclusion

Data availability statement

Author contributions

Funding

Conflict of interest

Generative AI statement

Publisher’s note

References

94% of researchers rate our articles as excellent or good

Expected values of topological descriptors for possible kink chains of type ₂⊤2

2.1 Vertex and corresponding edge partitions of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

3 Topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

4 Graphical representation of numerical values of topological descriptors of ${}_{2}⊤_{2}^{1}$ , ${}_{2}⊤_{2}^{2}$ and ${}_{2}⊤_{2}^{3}$

5 Expected values of topological descriptors of kink chains of type ${}_{2}⊤_{2}$

5.1 Results at ${(\overset{́}{κ} - 1)}^{t h}$ stage

5.2 Analytical expressions at ${(k - 1)}^{t h}$ stage

5.3 A comparison of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage

5.4 Graphical representation of expected values of topological descriptors at ${(\overset{́}{κ} - 1)}^{t h}$ stage