Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network

The Zagreb connection indices are the known topological descriptors of the graphs that are constructed from the connection cardinality (degree of given nodes lying at a distance 2) presented in 1972 to determine the total electron energy of the alternate hydrocarbons. For a long time, these connection indices did not receive much research attention. Ali and Trinajstić [Mol. Inform. 37, Art. No. 1800008, 2018] examined the Zagreb connection indices and found that they compared to basic Zagreb indices and that they provide a finer value for the correlation coefficient for the 13 physico-chemical characteristics of the octane isomers. This article acquires the formulae of expected values of the first Zagreb connection index of a random cyclooctatetraene chain, a random polyphenyls chain, and a random chain network with l number of octagons, hexagons, and pentagons, respectively. The article presents extreme and average values of all the above random chains concerning a set of special chains, including the meta-chain, the ortho-chain, and the para-chain.

1 Introduction

Graph theory is vital to various disciplines, including the chemical and biological sciences. One of the objectives of chemical graph theory is its primary and significant role in studying physico-chemical reactions and biological activities and pointing out the structural properties of molecular graphs, etc., Topological descriptors have played a significant role in achieving the desired properties of molecular graphs. Topological descriptors are molecular structural invariants that theoretically and mathematically explain the connectivity characteristics of nano-materials and chemical compounds. Therefore, topological indices produce sharper approaches to measuring their behavior and characteristics.

For 20 years, hydrocarbons and their derivatives have received attention from researchers because these compounds only have two members, carbon and hydrogen. We can acquire various types of hydrocarbon derivatives by replacing their molecular hydrogen atoms with different types of other atomic groups. A large number of valuable hydrocarbons are available in plants and some valuable characteristics of hydrocarbons are important to chemical raw materials and fuel.

Throughout this article, the vertex and edge sets of a graph $\mathcal{H}$ are represented as $V(\mathcal{H})$ and $E(\mathcal{H})$, respectively. We denote the degree of a vertex $\mathsf{v}\in V(\mathcal{H})$ by $d_{\mathcal{H}}(\mathsf{v})$, which is defined as the cardinality of edges joined with $\mathsf{v}$. Let the order and size of $\mathcal{H}$ be n and m, respectively. The l-degree of a given vertex $\mathsf{v}\in V(\mathcal{H})$, presented by $d_l(\mathsf{v})$, is the cardinality of set of vertices of $V(\mathcal{H})$ whose distance from $\mathsf{v}$ is l, where $d_1(\mathsf{v})=d_{\mathcal{H}}(\mathsf{v})$ and $d_2(\mathsf{v})={\tau }_{\mathsf{v}}$ [this is known as the connection number of $\mathsf{v}$ (Todeschini and Consonni, 2000)].

Suppose that $\mathbb{Z}$ is a collection of all connected simple graphs. There is a function $\mathcal{P}:\mathbb{Z}\to {\mathbb{R}}^{+}$ that describes a topological invariant if for any two isomorphic members ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ of $\mathbb{Z}$, we have $\mathcal{P}({\mathcal{M}}_1)=\mathcal{P}({\mathcal{M}}_2)$. Thousands of degree and distance-related topological invariants have been proposed, but some are better known because of their high predictive power for many characteristics like density, boiling point, molecular weight, refractive index, etc., Topological invariants have so many implementations in numerous areas of sciences such as drug discovery, physico-chemical research, toxicology, biology, and chemistry. To date, topological indices are the most notable field of graphical research. For more discussion on numerous invariants, we refer readers to studies by (Gutman, 2013; Akhter et al., 2016; Akhter and Imran, 2016; Akhter et al., 2017; Akhter et al., 2018; Akhter et al., 2020).

The Zagreb indices are the most notable invariants, and they have many valuable applications in chemistry. In 1972 Gutman and Trinajstić (Gutman and Trinajstić, 1972) established the first vertex degree dependent Zagreb index of a graph $\mathcal{H}$. Two renowned Zagreb indices of a graph $\mathcal{H}$ can be described in the following manner:

$\begin{array}{l}\hfill {\mathcal{M}}_1\left(\mathcal{H}\right)=\sum _{\mathsf{x}\in V\left(\mathcal{H}\right)}{d}_{\mathcal{H}}^2\left(\mathsf{x}\right),\\ \hfill {\mathcal{M}}_2\left(\mathcal{H}\right)=\sum _{\mathsf{x}\mathsf{y}\in E\left(\mathcal{H}\right)}{d}_{\mathcal{H}}\left(\mathsf{x}\right)d_{\mathcal{H}}\left(\mathsf{y}\right).\end{array}$

Motivated by how influential they have become and the many important applications of primary Zagreb indices, Naji et al. (Soner and Naji, 2016; Gutman et al., 2017) presented the concept of Zagreb connection indices (leap Zagreb indices), constructed from the second degrees of the vertices of a graph $\mathcal{H}$. The first, second, and modified Zagreb connection indices of $\mathcal{H}$ can be defined as:

$\begin{array}{l}\hfill Z{C}_1\left(\mathcal{H}\right)=\sum _{\mathsf{y}\in V\left(\mathcal{H}\right)}{\tau }_{\mathsf{y}}^2,\\ \hfill Z{C}_2\left(\mathcal{H}\right)=\sum _{\mathsf{xy}\in E\left(\mathcal{H}\right)}{\tau }_{\mathsf{x}}{\tau }_{\mathsf{y}},\\ \hfill Z{C}_1^{*}\left(\mathcal{H}\right)=\sum _{\mathsf{x}\in V\left(\mathcal{H}\right)}{d}_{\mathcal{H}}\left(\mathsf{x}\right){\tau }_{\mathsf{x}}.\end{array}$

The chemical applications of ZC₁ were presented in (8), indicating that the given index has a wide co-relation with the physical characteristics of chemical compounds, for instance, boiling point, enthalpy of evaporation, entropy, acentric factor, and standard enthalpy of vaporization. Let f_l present the cardinality of the subset of vertices of $\mathcal{H}$ with connection number l. The next formula for the first Zagreb connection index is equal to the above definition.

$Z{C}_1\left(\mathcal{H}\right)=\sum _{0\le l\le \mathsf{n}-2}{f}_l\left(G\right)l^2.$(1.1)

Naji and Soner (2018), (Gutman et al., 2017) determined the leap Zagreb descriptors of some graph operations and families. Leap Zagreb indices are presented in a recently published survey (Gutman et al., 2020). In (39), the authors establish sharp bounds for the leap Zagreb indices of trees and unicyclic graphs and also determined the corresponding extremal graphs. For more studies on Zagreb connection indices, we refer the readers to (Ducoffe et al., 2018a; Ali and Trinajstić, 2018; Shao et al., 2018a; Basavanagoud and Chitra, 2018; Ducoffe et al., 2018b; Khalid et al., 2018; Manzoor et al., 2018; Du et al., 2019; Fatima et al., 2019; Tang et al., 2019; Ye et al., 2019; Raza, 2020a; Bao et al., 2020; Raza, 2020b; Cao et al., 2020; Naji et al., 2020; Raza, 2022).

Huang et al. (2014) determined the expected values for Kirchhoff indices of random polyphenyl and spiro chains. Ma et al. (2017), Yang and Zhang. (2012), and Qi et al. (2022) independently acquired the expected value of Wiener indices of random polyphenyl chain and random spiro chain. Zhang et al. (2020) have provided expected values of the Schultz, Gutman, multiplicative degree-Kirchhoff, and additive degree-Kirchhoff indices of random polyphenylene chains. Raza and Imran. (2021) obtained expected values of modified second Zagreb, symmetric difference, inverse symmetric, and augmented Zagreb indices in random cyclooctane chains. Zhang et al. (2021) established the formulae for expected values of Sombor indices of a general random chain. Recently, many studies have explored the expected values of different topological indices. For further information, we refer readers to the following studies (Raza, 2020b; Fang et al., 2021; Raza, 2021; Jahanbanni, 2022; Raza et al., 2022).

Motivated by the above research, the present study determined the explicit formulae for expected values of the first Zagreb connection index of the random cyclooctatetraene chain, random polyphenyls chain, and random chain network with l octagons, hexagons, and pentagons, respectively. Moreover, we examined the average and extreme values of the Zagreb connection index among all the above-mentioned random chains corresponding to their set.

2 The first Zagreb connection index of random cyclooctatetraene chain

Cyclooctatetraene, having chemical formula C₈H₈, is an organic compound whose full name is ‘1, 3, 5, 7 − cyclooctene. Its structure is a cyclic polyolefin-like benzene, but it is not aromatic, see (Willis et al., 1952; Mathews and Lipscomb, 1959; Traetteberg et al., 1970). It has the same chemical characteristics as unsaturated hydrocarbons and is easy to construct explosive organic peroxides, (Milas and NolanPetrus, 1958; Donald and Whitehead, 1969; Garavelli et al., 2002; Schwamm et al., 2019).

Spiro compounds are valuable types of cycloaltanes in organic chemistry. A spiro union is a join of two rings that have a common atom between both rings, and a join of a direct union among the rings is known as a free spiro union in spiro compounds. In a cyclooctatylene chain, octagons are joined by cut vertices or cut edges. A random cyclooctatetraene chain COC_l, has l octagons, and can be constructed by a cyclooctatetraene chain COC_l−1 with l−1 octagons attached to a new octagon G_l by a bridge (see Figure 1).

FIGURE 1

FIGURE 1. A random cyclooctatetraene chain COC_l.

The COC_l is a cyclooctatetraene chain with l ≥ 2 having G₁, G₂, … , G_l octagons. The new octagon can be joined by four different schemes, which give the local orderings. We use these as $CO{C}_l^1$, $CO{C}_l^2$, $CO{C}_l^3$, $CO{C}_l^4$(see Figure 2).

FIGURE 2

FIGURE 2. The four types of local arrangements of octagons $CO{C}_l^1$, $CO{C}_l^2$, $CO{C}_l^3$ and $CO{C}_l^4$.

A random cyclooctatetraene chain COC_l(k₁, k₂, k₃) is a cyclooctatetraene chain constructed by step-by-step attachment of new octagons. At every step p = 2, 3, … , l a random choice is constructed from one of the four possible chains:

1 $CO{C}_{p-1}\to CO{C}_p^1$ with probability k₁,

2 $CO{C}_{p-1}\to CO{C}_p^2$ with probability k₂,

3 $CO{C}_{p-1}\to CO{C}_p^3$ with probability k₃,

4 $CO{C}_{p-1}\to CO{C}_p^4$ with probability k₄ = 1 − k₁−k₂−k₃,

Where all the given probabilities are constant. In this section, we will discuss the expected value for the first Zagreb connection index among random cyclooctatetraene chains with l octagons.

3 The first Zagreb connection index of a random polyphenyls chain

Polyphenyls showed a molecular graph corresponding to a type of macrocyclic aromatic hydrocarbons, and these molecular graphs of polyphenyls construct a polyphenyl structure. Polyphenyls and their derivatives have applications in drug synthesis, organic synthesis, heat exchangers, etc., and have received attention from chemists. A random polyphenyl chain PPC_l with l hexagons can be constructed by a polyphenyl chain PPC_l−1 using l−1 hexagons attached to a new hexagon G_l by a bridge (see Figure 3).

FIGURE 3

FIGURE 3. The three types of local arrangements of hexagons $PP{C}_l^1$, $PP{C}_l^2$, and $PP{C}_l^3$.

The PCC_l will be a polyphenyl chain with l ≥ 2 having G₁, G₂, … , G_l hexagons. PPC_l is the meta-chain M_l, the ortho-chain $O_l^1$ and the para-chain L_l. The new hexagon can be joined in three arrangements, which construct the local orderings. We use these as $PP{C}_l^1$, $PP{C}_l^2$, $PP{C}_l^3$ (see Figure 4).

FIGURE 4

FIGURE 4. A random polyphenyl chain PPC_l.

A random polyphenyl chain PPC_l(k₁, k₂) is a polyphenyl chain constructed by step-by-step attachment of new hexagons. At every step p = 2, 3, … , l, a random choice construct one of the three possible chains:

1 $PP{C}_{p-1}\to PP{C}_p^1$ with probability k₁,

2 $PP{C}_{p-1}\to PP{C}_p^2$ with probability k₂,

3 $PP{C}_{p-1}\to PP{C}_p^3$ with probability k₃ = 1 − k₁−k₂,

Where all the given probabilities are constant. In this section, we discuss the expected value for the first Zagreb connection index of the random polyphenyl chain with l hexagons.

4 The first Zagreb connection index of random chain network PG_l

The random chain networks PG_l with l pentagons can be constructed by PG_l−1 having l−1 pentagons attached to a new pentagon H_l by a bridge (see Figure 5).

FIGURE 5

FIGURE 5. A random chain networks PG_l.

The PG_l will be a random chain network with l ≥ 2, and H₁, H₂, … , H_l pentagons. For l ≥ 3, there are two ways to attach pentagons at the end and get $P{G}_l^1$ and $P{G}_l^2$, (see Figure 6). For such a random chain network, any step for q = 2, 3, 4, … , l can be constructed by two possible chains with given probabilities k₁ and k₂, respectively:

1 $P{G}_{q-1}\to P{G}_q^1$ with probability k₁,

2 $P{G}_{q-1}\to P{G}_q^1$ with probability k₂ = 1 − k₁,

FIGURE 6

FIGURE 6. The two types of local arrangements of pentagons $P{G}_l^1$ and $P{G}_l^2$.

Where all the given probabilities are constant.

This section discusses the expected value for the first Zagreb connection index of the random chain network with l pentagons. The proof of Theorem 4.1 is the same as the proofs of Theorem 2.1 and Theorem 3.1; therefore, we omit it here.

5 The average values for the first Zagreb connection index

This section finds the average values for the first Zagreb connection index concerning the sets of all cyclooctatetraene chains with l octagons, polyphenyl chains with l hexagons, and chain networks with l pentagons. Let ${\mathbb{G}}_l$, ${\mathbb{R}}_l$ and ${\mathbb{Q}}_l$ be the sets of all cyclooctatetraene chains, polyphenyl chains, and random chain network, respectively. The average values for the first Zagreb connection index for the sets ${\mathbb{G}}_l$, ${\mathbb{R}}_l$ and ${\mathbb{Q}}_l$ are given below:

$\begin{array}{ll}\hfill & Z{C}_1^{\mathit{avg}}\left({\mathbb{G}}_l\right)=\frac{1}{{\mathbb{G}}_l}\sum _{\mathcal{H}\in {\mathbb{G}}_l}Z{C}_1\left(\mathcal{H}\right),\hfill \\ \hfill & Z{C}_1^{\mathit{avg}}\left({\mathbb{R}}_l\right)=\frac{1}{{\mathbb{R}}_l}\sum _{\mathcal{H}\in {\mathbb{R}}_l}Z{C}_1\left(\mathcal{H}\right),\hfill \\ \hfill & Z{C}_1^{\mathit{avg}}\left({\mathbb{Q}}_l\right)=\frac{1}{{\mathbb{Q}}_l}\sum _{\mathcal{H}\in {\mathbb{Q}}_l}Z{C}_1\left(\mathcal{H}\right).\hfill \end{array}$

The average value concerning sets ${\mathbb{G}}_l$, ${\mathbb{R}}_l$, and ${\mathbb{Q}}_l$ are expected values for the first Zagreb connection index of the random chains. From Theorem 2.1, Theorem 3.1 and Theorem 4.1, we have.

6 Conclusion

This study computed the expected values of the first Zagreb connection index in a random cyclooctatetraene chain, random polyphenyls chain, and random chain network with l, octagons, hexagons, and pentagons, respectively. It has discussed the maximum chain and the minimum chain of the COC_l, PPC_l, and PG_l, respectively, concerning the expected values of these chains. The average values discussed in all of the above are considered random chains for unique chains.

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

Investigation: ZR, SA, and YS: Writing: ZR, SA, and YS; Review: ZR, SA, and YS.

Funding

This research was funded by the University of Sharjah.

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.

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.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fchem.2022.1067874/full#supplementary-material

References

Akhter, S., Imran, M., Farahan, M. R., and Javaid, I. (2017). On topological properties of hexagonal and silicate networks. Hac. J. Math. Stat. 48 (3), 1–13. doi:10.15672/hjms.2017.541

CrossRef Full Text | Google Scholar

Akhter, S., Imran, M., Gao, W., and Farahani, M. R. (2018). On topological indices of honeycomb networks and graphene networks. Hac. J. Math. Stat. 47 (1), 19–35. doi:10.15672/hjms.2017.464

CrossRef Full Text | Google Scholar

Akhter, S., Imran, M., and Iqbal, Z. (2020). Mostar indices of SiO₂ nanostructures and melem chain nanostructures. Int. J. Quantum Chem. 121 (5). doi:10.1002/qua.26520

CrossRef Full Text | Google Scholar

Akhter, S., and Imran, M. (2016). On molecular topological properties of benzenoid structures. Can. J. Chem. 94 (8), 687–698. doi:10.1139/cjc-2016-0032

CrossRef Full Text | Google Scholar

Akhter, S., Imran, M., and Raza, Z. (2016). On the general sum-connectivity index and general Randić index of cacti. J. Inequal. Appl. 2016 (1), 300–309. doi:10.1186/s13660-016-1250-6

CrossRef Full Text | Google Scholar

Ali, A., and Trinajstić, N. (2018). A novel/old modification of the first Zagreb index. Mol. Inf. 37 (6–7), 1800008. doi:10.1002/minf.201800008

PubMed Abstract | CrossRef Full Text | Google Scholar

Bao, L. J., Raza, Z., and Javaid, M. (2020). Zagreb connection numbers for cellular neural networks. Discrete Dyn. Nat. Soc. 2020, 1–8. doi:10.1155/2020/8038304

CrossRef Full Text | Google Scholar

Basavanagoud, B., and Chitra, E. (2018). On the leap Zagreb indices of generalized xyz-point-line transformation graphs T_xyz(G), when z = 1. Int. J. Math. Comb. 2 (44–66).

Google Scholar

Basavanagoud, B., and Jakkannavar, P. (2018). Computing the first leap Zagreb index of some nanostructures. Inter. J. Appl. Math. 6 (2–B), 141–150.

Google Scholar

Cao, J., Ali, U., Javaid, M., and Huang, C. (2020). Zagreb connection indices of molecular graphs based on operations. Complexity 2020, 1–15. doi:10.1155/2020/7385682

CrossRef Full Text | Google Scholar

Donald, H. L., and Whitehead, M. A. (1969). Molecular geometry and bond energy. III. cyclooctatetraene and related compounds. J. Am. Chem. Soc. 91 (2), 238–242. doi:10.1021/ja01030a003

CrossRef Full Text | Google Scholar

Du, Z., Ali, A., and Trinajstić, N. (2019). Alkanes with the first three maximal/minimal modified first Zagreb connection indices. Mol. Inf. 38 (4), 1800116. doi:10.1002/minf.201800116

PubMed Abstract | CrossRef Full Text | Google Scholar

Ducoffe, G., Marinescu-Ghemeci, R., and Obreja, C. (2018). International symposium on symbolic and numeric algorithms for scientific computing (SYNASC). Timisoara, Rom. 141–148.

PubMed Abstract | Google Scholar

Ducoffe, G., Marinescu-Ghemeci, R., and Obreja, C. (2018). Proceedings of the 16th cologne-twente workshop on graphs and combinatorial optimization. Paris, Fr. CNAM 65–68.

PubMed Abstract | Google Scholar

Fang, X., You, L., and Liu, H. (2021). The expected values of Sombor indices in random hexagonal chains, phenylene chains, and Sombor indices of some chemical graphs. Int. J. Quantum. Chem. 121 (17). doi:10.1002/qua.26740

CrossRef Full Text | Google Scholar

Fatima, N., Bhatti, A. A., Ali, A., and Gao, W. (2019). Zagreb connection indices of two dendrimer nanostars. Acta Chem. Iasi 27 (1), 1–14. doi:10.2478/achi-2019-0001

CrossRef Full Text | Google Scholar

Garavelli, M., Bernardi, F., Cembran, A., Castaño, O., Frutos, L. M., Merchán, M., et al. (2002). Cyclooctatetraene computational photo- and thermal chemistry: A reactivity model for conjugated hydrocarbons. J. Am. Chem. Soc. 124 (46), 13770–13789. doi:10.1021/ja020741v

PubMed Abstract | CrossRef Full Text | Google Scholar

Gutman, I. (2013). Degree-based topological indices. Croat. Chem. Acta. 86, 351–361. doi:10.5562/cca2294

CrossRef Full Text | Google Scholar

Gutman, I., Milovanović, E., and Milovanović, I. (2020). Beyond the Zagreb indices. AKCE Int. J. Graphs Comb. 17 (1), 74–85. doi:10.1016/j.akcej.2018.05.002

CrossRef Full Text | Google Scholar

Gutman, I., Naji, A. M., and Soner, N. D. (2017). The first leap Zagreb index of some graph operations. Commun. Comb. Optim. 2 (2), 99–117.

Google Scholar

Gutman, I., and Trinajstić, N. (1972). Graph theory and molecular orbitals, Total π electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17 (4), 535–538. doi:10.1016/0009-2614(72)85099-1

CrossRef Full Text | Google Scholar

Huang, G. H., Kuang, M. J., and Deng, H. Y. (2014). The expected values of Kirchhoff indices in the random polyphenyl and spiro chains. ARS Math. Contem. 9 (2), 197–207. doi:10.26493/1855-3974.458.7b0

CrossRef Full Text | Google Scholar

Jahanbanni, A. (2022). The expected values of the first Zagreb and Randić indices in random polyphenyl chains. Polycycl. Aromat. Compd. 42 (4), 1851–1860. doi:10.1080/10406638.2020.1809472

CrossRef Full Text | Google Scholar

Khalid, S., Kok, J., and Ali, A. (2018). Zagreb connection indices of TiO₂ nanotubes. Chem. Bulg. J. Sci. Edu. 27 (1), 86–92.

Google Scholar

Ma, L., Bian, H., Liu, B. J., and Yu, H. Z. (2017). The expected values of the Wiener index in the random phenylene and spiro chains. ARS Comb. 130, 267–274.

Google Scholar

Manzoor, S., Fatima, N., Bhatti, A. A., and Ali, A. (2018). Zagreb connection indices of some nanostructures. Acta Chem. Iasi 26 (2), 169–180. doi:10.2478/achi-2018-0011

CrossRef Full Text | Google Scholar

Mathews, F. S., and Lipscomb, W. N. (1959). The structure of silver cyclooctatetraene nitrate. J. Phy. Chem. 63 (6), 845–850. doi:10.1021/j150576a017

CrossRef Full Text | Google Scholar

Milas, N., and NolanPetrus, J. (1958). Notes-ozonization of cyclooctatetraene. J. Org. Chem. 23 (4), 624–625. doi:10.1021/jo01098a611

CrossRef Full Text | Google Scholar

Naji, A. M., Davvaz, B., and Mahde, S. S. (2020). A study on some properties of leap graphs. Commun. Comb. Optim. 5 (1), 9–17.

Google Scholar

Naji, A. M., and Soner, N. D. (2018). The first leap Zagreb index of some graph operations. Int. J. Appl. Graph Theor. 2 (1), 7–18.

Google Scholar

Qi, J., Fang, M., and Geng, X. (2022). The expected value for the Wiener index in the random spiro chains. Polycycl. Aromat. Compd. 1–11.

PubMed Abstract | CrossRef Full Text | Google Scholar

Raza, Z., and Imran, M. (2021). Expected values of some molecular descriptors in random cyclooctane chains. Symmetry 13 (11), 2197. doi:10.3390/sym13112197

CrossRef Full Text | Google Scholar

Raza, Z. (2020). Leap Zagreb connection numbers for some networks models. J. Chem. 20 (6), 1407–1413. doi:10.22146/ijc.53393

CrossRef Full Text | Google Scholar

Raza, Z., Naz, K., and Ahmad, S. (2022). Expected values of molecular descriptors in random polyphenyl chains. Emerg. Sci. J. 6 (1), 151–165. doi:10.28991/esj-2022-06-01-012

CrossRef Full Text | Google Scholar

Raza, Z. (2020). The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains. Heliyon 6 (7), e04479. doi:10.1016/j.heliyon.2020.e04479

PubMed Abstract | CrossRef Full Text | Google Scholar

Raza, Z. (2021). The expected values of some indices in random phenylene chains. Eur. Phys. J. Plus 136 (11–15), 91. doi:10.1140/epjp/s13360-021-01082-y

CrossRef Full Text | Google Scholar

Raza, Z. (2022). Zagreb connection indices for some Benzenoid systems. Polycycl. Aromat. Compd. 42 (4), 1814–1827. doi:10.1080/10406638.2020.1809469

CrossRef Full Text | Google Scholar

Schwamm, R. J., Anker, M. D., Lein, M., and Coles, M. P. (2019). Reduction vs. Addition: The reaction of an aluminyl anion with 1, 3, 5, 7 − cyclooctatetraene. Angew. Chem. Int. Ed. 58 (5), 1489–1493. in English) 131. doi:10.1002/anie.201811675

PubMed Abstract | CrossRef Full Text | Google Scholar

Shao, Z., Gutman, I., and Li, Z. (2018). Leap Zagreb indices of trees and unicyclic graphs. Commun. Comb. Optim. 3 (2), 179–194.

Google Scholar

Shao, Z., Gutman, I., Li, Z., Wang, S., and Wu, P. (2018). Leap Zagreb indices of trees and unicyclic graphs. Commun. Comb. Optim. 3 (2), 179–194.

Google Scholar

Soner, N. D., and Naji, A. M. (2016). The k-distance neighborhood polynomial of a graph. Int. J. Math. Comput. Sci. 3 (9), 2359–2364.

Google Scholar

Tang, J. H., Ali, U., Javaid, M., and Shabbir, K. (2019). Zagreb connection indices of subdivision and semi-total point operations on graphs. J. Chem.

CrossRef Full Text | Google Scholar

Todeschini, R., and Consonni, V. (2000). Handbook of molecular descriptors. Weinheim: Wiley VCH.

Google Scholar

Traetteberg, M., Hagen, G., and Cyvin, S. J. (1970). IV. 1, 3, 5, 7-Cyclooctatetraene. Z. Für Naturforsch. B 25 (2), 134–138. doi:10.1515/znb-1970-0201

CrossRef Full Text | Google Scholar

Willis, B. P., George, C. P., and Kenneth, S. P. (1952). The structure of cyclooctatetraene. J. Am. Chem. Soc. 74 (13), 3437–3438. doi:10.1021/ja01133a524

CrossRef Full Text | Google Scholar

Yang, W., and Zhang, F. (2012). Wiener index in random polyphenyl chains. MATCH Commun. Math. Comp. Chem. 68, 371–376.

Google Scholar

Ye, A., Qureshi, M. I., Fahad, A., Aslam, A., Jamil, M. K., Zafar, A., et al. (2019). Zagreb connection number index of nanotubes and regular hexagonal lattice. Open Chem. 17 (1), 75–80. doi:10.1515/chem-2019-0007

CrossRef Full Text | Google Scholar

Zhang, L., Li, Q., Li, S., and Zhang, M. (2020). The expected values for the Schultz index, Gutman index, multiplicative degree Kirchhoff index, and additive degree Kirchhoff index of a random polyphenylene chain. Discrete Appl. Math. 282, 243–256. doi:10.1016/j.dam.2019.11.007

CrossRef Full Text | Google Scholar

Zhang, W., You, L., Liu, H., and Huang, Y. (2021). The expected values and variances for Sombor indices in a general random chain. App. Math. Comp. 411, 126521. doi:10.1016/j.amc.2021.126521

CrossRef Full Text | Google Scholar