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ORIGINAL RESEARCH article

Front. Chem., 09 July 2021
Sec. Theoretical and Computational Chemistry
This article is part of the Research Topic Frontiers in Chemistry - Rising Stars: Asia View all 13 articles

Edge Mostar Indices of Cacti Graph With Fixed Cycles

  • 1Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan
  • 2School of Natural Science, National University of Science and Technology, Islamabad, Pakistan

Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph , the edge Mostar invariant is described as Moe()=gxE()|m(g)m(x)|, where m(g)(or m(x)) is the number of edges of lying closer to vertex g (or x) than to vertex x (or g). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and n vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in (n,s), where s is the number of cycles.

1 Introduction

Let =(V(),E()) be a simple, undirected, and connected graph with the vertex set V() and the edge set E(). The degree of gV(), represented as deg(g), is described as the number of edges directly linked with g. The neighbors of a vertex g in is the set of all of its adjacent vertices in . For g,xV(), the number of edges in the shortest path between two vertices g and x is called the distance between them and is expressed as d(g,x). A pendent vertex p in is a vertex with degree one, and an edge having one pendent vertex as one of its end vertices is called a pendent edge. The set of all pendent vertices of is represented as P, and the set of all pendent vertices adjacent to a fix vertex g is represented as P(g). An edge in is presented as a cut edge if, by deleting that edge, the graph is converted into exactly two components. Let Pn, Cn, and Sn be used for the representation of the path, the cycle, and the star with order n.

In the fields of chemical sciences, mathematical chemistry, chemical graph theory, and pharmaceutical science, topological invariants are of significant importance because of their definitional use. The physicochemical properties of chemical structures can be forecasted by using topological invariants. A numerical value related to biological activity, chemical reactivity, and physical properties of chemical structures is known as a topological invariant. Topological invariants are mainly separated into different manners like degree, distance, eccentricity, and spectrum. A distance-based invariant is a topological invariant based on the distance between the vertices or edges of a given graph. The Wiener invariant (Wiener, 1947) is the most significant oldest topological invariant that belongs to distance-based invariants, and the Harary invariant (Mihalić and Trinajstić, 1992) and the Balaban invariant (Zhou and Trinajstić, 2008) also belong to distance-based invariants. Degree-based invariants are another well-studied group of invariants. The first degree-based invariant was introduced as the Randić invariant (Randic, 1975). A rich theory of distance- and degree-based invariants is mentioned in (Li and Shi, 2008; Gutman, 2013; Knor et al., 2014; Knor et al., 2015). The recently introduced Mostar invariant (Došlić et al., 2018) belongs to bound additive invariants as they capture the relevant properties of a graph by summing up the contributions of individual edges (Vukičević and Gašperov, 2010; Vukičević, 2011). Peripherality is one such property that could be of interest. An edge is a peripheral edge if there are many more vertices closer to one of its end vertices than to the other one. In short, for an edge gx in , the greatest value of absolute difference of the cardinality of vertices closer to g than to x, presented by n(g), and the cardinality of vertices closer to x than to g, denoted by n(x), indicates a peripheral position of gx in . The Mostar invariant of a graph is defined as follows:

Mov()=e=gxE()|n(g)n(x)|,(1)

and this represents a global measure of peripherality of a graph . Došlić et al. (2018) determined the Mostar invariant of the benzenoid system. Tratnik proved that the Mostar invariant of the weighted graph can be deduced in the form of the Mostar invariant of quotient graphs (Tratnik, 2019). Arockiaraj et al. (2019) introduced the edge Mostar invariant as follows:

Moe()=e=gxE()|m(g)m(x)|,(2)

where m(g)(or m(x)) is the cardinality of edges closer to g (or x) than to x (or g).Akhter et al. (2021) computed the Mostar indices for the molecular graphs of SiO2 layer structures and the melem chain with the help of the cut method. Liu et al. (2020) found the extremal values of the edge Mostar invariant of cacti graphs. Imran et al. (2020) found the edge Mostar invariant of chemical structures and nanostructures using graph operations. Arockiaraj et al. (2020) calculated the weighted Mostar indices of molecular peripheral shapes with applications in graphene, graphyne, and graphdiyne nanoribbons. Liu et al. (2020) determined the maximum edge Mostar index of cacti graphs with the following given conditions.

Theorem 1.1.LetG(n,s)be a connected graph:

ifn10andn<4s, thenMoe(G)2n28n+(244n)swith equality if and only ifGGn(3,3,3,34sn,4,4,44n3s),

ifn10andn4s, thenMoe(G)n2n12swith equality if and only ifGGn(4,4,4),

ifn=9, thenMoe(G)7=7212swith equality if and only ifGG9, and

ifn<9, thenMoe(G)n2n(n+3)swith equality if and only ifGGn(3,3,3,3).

Liu et al. (2020) determined the second maximum edge Mostar index of cacti graphs with the following given conditions.

Theorem 1.2.LetG(n,s)\0(n,s)withn10andn4s>0:

Moe(G)8912sforn=10with equality if and only ifGG(3,4,4,44s1),

Moe(G)10812sforn=11with equality if and only ifGG(3,4,4,44s1), and

Moe(G)n2n12s2with equality if and only ifGG1(n,s).

For more results related to Mostar and edge Mostar invariants, see (Hayat and Zhou, 2019a; Akhter, 2019; Tepeh, 2019; Akhter et al., 2020; Dehgardi and Azari, 2020; Deng and Li, 2020; Ghorbani et al., 2020; Huang et al., 2020; Deng and Li, 2021a; Deng and Li, 2021b). A connected graph is a cactus if all its blocks are either edges or cycles, that is, any two of its cycles have at most one common vertex. Until now, many results in chemistry and graph theory related to the cacti have been acquired. The first three smallest Gutman invariants among the cacti have been determined by Chen (2016). Using the Zagreb invariants, Li et al. (2012) found the upper and lower bounds of the cacti. The bounds of the Harary invariant related to cacti have been found by Wang and Kang (2013). The extremal cacti having the greatest hyper-Wiener invariant have been characterized by Wang and Tan (2015). The extremal graphs with the greatest and smallest vertex PI invariants among all cacti with a fixed number of vertices have been determined by Wang et al. (2016). The sharp upper bound of the Mostar invariant for cacti of order n with s cycles has been given by Hayat and Zhou (2019b), and they also found the greatest Mostar invariant for all n-vertex cacti. For more results related to cacti graphs, see (Liu et al., 2016; Wang and Wei, 2016; Wang, 2017). Motivated by the results of chemical invariants and their applications, it may be interesting to characterize the cacti with the greatest and smallest edge Mostar invariants for some fixed parameters. In this study, we consider the cacti with a fixed number of cycles and find the greatest edge Mostar invariant for all the n-vertex cacti. In the end, we give a sharp upper bound of the edge Mostar invariant for these cacti.

2 Main Results

Let (n) be the set of all cacti graphs of order n2 and (n,s) be the set of all cacti graphs of order n2 with the number of cycles s. Let ^(n,s)(n,s) be the n-vertex cactus, for n3s+2, s2 and for n9, s=1, consisting of s number of C4 and n3s1 pendent edges such that every c4 and pendent edge has exactly one vertex in common (see Figure 1).

FIGURE 1
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FIGURE 1. Graph ^(n,s), for n3s+2, s2 and for n9, s=1.

In this section, we derive the greatest value of cacti graphs for the edge Mostar invariant. First of all, some basic lemmas are proved so that the main result can be proved easily.

Proposition 2.1.(Imran et al., 2020) The edge Mostar invariant of a pathPnand a cycleCnwith n vertices isMoe(Pn)=[(n1)22]andMoe(Cn)=0, respectively. In Lemma 2.1, we establish a graph G2 by converting a cut edge uv into a pendent edge uw in G1, such that the new graph G2 has a greater edge Mostar invariant.

Lemma 2.1:Consider two connected graphsH1andH2such that they are connected to each other by an edgeuv, whereuV(H1)andvV(H2), and acquired the graphG1. Now, we construct the graphG2by deleting the cut edgeuvand attaching a pendent edgeuwat vertex u inG1 (seeFigure 2). ThenMoe(G1)<Moe(G2).

FIGURE 2
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FIGURE 2. Graphs G1 and G2 of Lemma 2.1.

Proof: Let H1 and H2 be the subgraphs of G1, as shown in Figure 2. By the construction of G2, the number of closer edges of the end vertices of a fixed edge of H1 and H2 in G1 remains the same in G2, respectively. Therefore, for an edge gxE(Hl), where l{1,2}, we have the following:

mG1(g)=mG2(g),mG1(x)=mG2(x)(3)

For the cut edge uv in G1 and the pendent edge uw in G2, we have the following:

mG1(u)=|E(H1)|,mG1(v)=|E(H2)|,
mG2(u)=|E(H1)|+|E(H2)|,mG2(w)=0.(4)

Using the definition of the edge Mostar invariant and substituting the values from Eqs 3, 4 , we acquire the following:

Moe(G1)Moe(G2)=|mG1(u)mG1(v)|+l=12gxE(Hl)|mG1(g)mG1(x)||mG2(u)mG2(w)|l=12gxE(Hl)|mG2(g)mG2(x)|=||E(H1)|+|E(H2)||+l=12gxE(Hl)|mG1(g)mG1(x)|||E(H2)||E(H1)||l=12gxE(Hl)|mG1(g)mG1(x)|=E(H1)||E(H2)E(H1)|+|E(H2).

There are two cases:

1. if |E(H1)|>|E(H2)|, then we get |E(H1)||E(H2)||E(H1)||E(H2)|=2|E(H2)|<0, and

2. if |E(H1)|<|E(H2)|, then we get |E(H1)|+|E(H2)||E(H1)||E(H2)|=2|E(H1)|<0.

In either case, we acquire Moe(G1)Moe(G2)<0.This completes the proof. ∎Next, we establish a new G2 graph from G1 by moving all pendent edges, all C4 cycles, and all C3 cycles from different vertices of a fixed cycle Cs to a unique vertex, such that the new graph has a larger edge Mostar invariant.

Lemma 2.2:LetGbe a cyclic graph constructed by attachingri, forri0, number of pendent vertices,ti, forti0, number ofC4cycles andmi, formi0, number ofC3cycles, at the verticesvi, for1is1, ofCs, wheres3. Consider a graph H having a common vertexvV(H)withGand present it byG1. We constructG2fromG1by removing all the pendent vertices,C4’s, andC3’s ofGand attaching them at v (seeFigure 3). Then, we haveMoe(G1)<Moe(G2).

FIGURE 3
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FIGURE 3. Graphs G1 and G2 of Lemma 2.2.

Proof:Suppose that the vertices of Cs are v0(=v),v1,v2,,vs1 and there are ri number of pendent edges, ti number of C4 cycles, and mi number of C3 cycles rooted at vi, for 1is1, in G1. By the construction of G2, the number of closer edges of the end vertices of a fixed edge of H in G1 remains the same in G2. Therefore, for any edge u1u2E(H), we have the following:

mG1(u1)=mG2(u1),mG1(u2)=mG2(u2).(5)

For the pendent edges viu rooted on vi, for 1is1 and uP(G), in G1, we have the following:

mG1(vi)=|E(H)|+|E(G)|1,mG1(u)=0=mG2(u)mG2(v=v0)=|E(H)|+|E(G)|1.(6)

For every C4 cycle rooted on a fixed vertex vi, for 1is1, the edge set is {w0w1,w1w2,w2w3,w3w0}, and then, there are the following cases:

1. For wiwi+1, i=0,1, we have mG1(wi)=|E(H)|+|E(G)|3=mG2(wi) and mG1(wi+1)=1=mG2(wi+1).

2. For w2w3, we have mG1(w3)=|E(H)|+|E(G)|3=mG2(w3) and mG1(w2)=1=mG2(w2).

3. For w0w3, we have mG1(w0)=|E(H)|+|E(G)|3=mG2(w0) and mG1(w3)=1=mG2(w3).

For every C3 cycle rooted on a fixed vertex vi, for 1is1, the edge set is {g0g1,g1g2,g2g3,g3g0}, and then, there are the following cases:

1. For g0g1, we have mG1(g0)=|E(H)|+|E(G)|2=mG2(g0) and mG1(g1)=1=mG2(g1).

2. For g1g2, we have mG1(g3)=mG1(g2) and mG2(g3)=mG2(g2).

3. For g0g2, we have mG1(g0)=|E(H)|+|E(G)|2=mG2(g0) and mG1(g2)=1=mG2(g2).

Suppose Cs is an even cycle; then there are the following cases:

1. For v0v1, we have mG1(v0)=|E(H)|+s21+p=s2+1s1(rp+tp+mp) and mG1(v1)=s21+p=1s2(rp+tp+mp).

2. For vivi+1, where 1is21, we have mG1(vi)=|E(H)|+s21+p=s2+i+1s1(rp+tp+mp)+p=1i(rp+tp+mp) and mG1(vi+1)=s21+p=i+1s2+i(rp+tp+mp).

3. For vivi+1, where s2is2, we have mG1(vi)=s21+p=i(s21)i(rp+tp+mp) and mG1(vi+1)=|E(H)|+s21+p=i+1s1(rp+tp+mp)+p=1is2(rp+tp+mp).

4. For v0vs1, we have mG1(v0)=|E(H)|+s21+p=1s21(rp+tp+mp) and mG1(vs1)=s21+p=s2s1(rp+tp+mp).

5. For vivi+1, where 0is21, we have mG2(vi)=|E(H)|+s21+p=1s1(rp+tp+mp) and mG2(vi+1)=s21.

6. For vivi+1, where s2is2, we have mG2(vi)=s21 and mG2(vi+1)=|E(H)|+s21+p=1s1(rp+tp+mp).

7. For v0vs1, we have mG2(v0)=|E(H)|+s21+p=1s1(rp+tp+mp) and mG2(vs1)=s21.

Substituting the values from Eqs 5, 6 and the information from all the cases above in the definition of the edge Mostar invariant, we acquire the following:

Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+viuE(G),uPG(vi)|mG1(vi)mG1(u)|+i=03wiwi+1E(C4)|mG1(wi)mG1(wi+1)|+|mG1(w0)mG1(w3)|+i=02gigi+1E(C3)|mG1(gi)mG1(gi+1)|+|mG1(g0)mG1(g2)|+i=0s2|mG1(vi)mG1(vi+1)|+|mG1(v0)mG1(vs1)|u1u2E(H)|mG2(u1)mG2(u2)|vuE(G),uPG(vi)|mG2(u)mG2(v)|i=03wiwi+1E(C4)|mG2(wi)mG2(wi+1)||mG2(w0)mG2(w3)|i=02gigi+1E(C3)|mG2(gi)mG2(gi+1)||mG2(g0)mG2(g2)|i=0s2|mG2(vi)mG2(vi+1)||mG2(v0)mG2(vs1)|=u1u2E(H)|m1(u1)m1(u2)|+r||E(H)|+|E(G)|1|+4t||E(H)|+|E(G)|4|+2m||E(H)|+|E(G)|3|+||E(H)|+s21+p=s2+i+1s1(rp+tp+mp)+s21s2+1p=i+1s2+i(rp+tp+mp)|+i=s2s2|p=i(s21)i(rp+tp+mp)+s21s2+1p=i+1s1(rp+tp+mp)|E(H)|p=1i(rp+tp+mp)|+||E(H)|+s21.+p=s2+1s1(rp+tp+mp)s2+1p=1s2(rp+tp+mp)|u1u2E(H)|m1(u1)m1(u2)|r||E(H)|+|E(G)|1|4t||E(H)|+|E(G)|4|2m||E(H)|+|E(G)|3|i=0s21||E(H)|+s21+p=1s1(rp+tp+mp)s2+1|i=s2s2|+s21|E(H)|s2+1p=1s1(rp+tp+mp)|||E(H)|+s21+p=1s1(rp+tp+mp)s2+1||E(H)|+p=1s1(rp+tp+mp)+i=1s21(|E(H)|+p=1s1(rp+tp+mp))+i=s2s2(|E(H)|+p=1s1(rp+tp+mp))+|E(H)|+p=1s1(rp+tp+mp)i=0s21(|E(H)|+p=1s1(rp+tp+mp))i=s2s2(|E(H)|+p=1s1(rp+tp+mp))(|E(H)|+p=1s1(rp+tp+mp))|E(H)|+r+t+m+i=1s2(|E(H)|+r+t+m)i=0s2(|E(H)|+r+t+m)|E(H)|+r+t+m+(s2)(|E(H)|+r+t+m)(s1)(|E(H)|+r+t+m)0.

The proof for an odd cycle Cs is similar to that above; therefore, we omit it here.This completes the proof. ∎In Lemma 2.3, we establish a new graph G2 from a given graph G1 by replacing Cq with C4 and attaching q4 pendent edges in G1 such that the new graph has a greater edge Mostar invariant.

Lemma 2.3:Consider a graph H having a common vertexvV(H)withCqsuch thatdegH(v)3andq5, and denote it asG1. LetG2be the graph acquired fromG1by replacingCqwithC4and attachingq4pendent edges atvV(H)(seeFigure 4). Then, we haveMoe(G1)Moe(G2).

FIGURE 4
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FIGURE 4. Graphs G1 and G2 of Lemma 2.3.

Proof: Let H be a subgraph of G1 and the vertices of Cq be v0(=v),v1,v2,,vq1, as shown in Figure 4. By the construction of G2, the number of closer edges of the end vertices of a fixed edge of H in G1 remains the same in G2. Therefore, for any edge u1u2E(H), we have the following:

mG1(u1)=mG2(u1),mG1(u2)=mG2(u2).(7)

Suppose q is even; then there are three cases:

1. For vivi+1, where 0iq21, we have mG1(vi)=|E(H)|+q21 and mG1(vi+1)=q21.

2. For vivi+1, where q2iq2, we have mG1(vi)=q21 and mG1(vi+1)=|E(H)|+q21.

3. For v0vq1, we have mG1(v0)=|E(H)|+q21 and mG1(vq1)=q21.

Suppose q is odd; then there are three cases:

1. For vivi+1, where 0iq21, we have mG1(vi)=|E(H)|+q12 and mG1(vi+1)=q12.

2. For vivi+1, where q2iq2, we have mG1(vi)=q12 and mG1(vi+1)=|E(H)|+q12.

3. For v0vq1, we have mG1(v0)=|E(H)|+q12 and mG1(vq1)=q12.

In G2, for any pendent edge vvi, where 4iq1, rooted at v, we have the following:

mG2(v)=|E(H)|+q1,mG2(vi)=0.(8)

For v0v1,v1v2,v2v3,v3v0 in G2, there are the following cases:

1. For vivi+1, i=0,1, we have mG2(vi)=|E(H)|+q3 and mG2(vi+1)=1.

2. For v2v3, we have mG2(v3)=|E(H)|+q3 and mG2(v2)=1.

3. For v0v3, we have mG2(v0)=|E(H)|+q3 and mG2(v3)=1.

Case 1:When q is even, using the definition of the edge Mostar invariant and substituting the values from Eqs 7, 8 and the cases above, we get the following:

Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q2|mG1(vi)mG1(vi+1)|+|mG1(v0)mG1(vq1)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1|mG2(v)mG2(vi)||mG2(v0)mG2(v1)||mG2(v1)mG2(v2)||mG2(v2)mG2(v3)||mG2(v3)mG2(v0)|=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21||E(H)|+q21(q21)|+i=q2q2|q21(|E(H)|+q21)|+||E(H)|+q21(q21)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1||E(H)|+q10)|4||E(H)|+q31|=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21|E(H)|+i=q2q2|E(H)|+|E(H)|u1u2E(H)|mG1(u1)mG1(u2)|i=4q1||E(H)|+q1|4||E(H)|+q4|q|E(H)|(q4)|E(H)|(q4)q+(q4)4|E(H)|4q+16q2+q+12<0.

Case 2:When q is odd, using the definition of the edge Mostar invariant and substituting the values from Eqs 7, 8 and the cases above, we get the following:

Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q2|mG1(vi)mG1(vi+1)|+|mG1(v0)mG1(vvq1)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1|mG2(v)mG2(vi)||mG2(v0)mG2(v1)||mG2(v1)mG2(v2)||mG2(v2)mG2(v3)||mG2(v3)mG2(v0)|,
Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21||E(H)|+q12(q12)|+i=q2q2|q12(|E(H)|+q12)|+||E(H)|+q12(q12)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1||E(H)|+q10)|4||E(H)|+q31|=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21|E(H)|+i=q2q2|E(H)|+|E(H)|u1u2E(H)|mG1(u1)mG1(u2)|i=4q1||E(H)|+q1|4||E(H)|+q4|q|E(H)|(q4)|E(H)|(q4)q+(q4)4|E(H)|4q+16q2+q+12<0.

This completes the proof. ∎

Lemma 2.4:Consider a graph H having a common vertexvV(H)withC3and at least one pendent edgevu, and this graph is presented asG1. LetG2be the graph obtained fromG1by replacingC3andvuwithC4(seeFigure 5). Then, we haveMoe(G1)<Moe(G2).

FIGURE 5
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FIGURE 5. Graphs G1. and G2 of Lemma 2.4.

Proof: By the construction of G2, the number of closer edges of the end vertices of a fixed edge of H in G1 remains the same in G2. Therefore, for any edge u1u2E(H), we have the following:

mG1(u1)=mG2(u1),mG1(u2)=mG2(u2).(9)

There are the following cases in G1:

1. For pendent edge uvE(G1), we have mG1(v)=|E(H)|+3 and mG1(u)=0.

2. For vu1E(C3), we have mG1(v)=|E(H)|+2 and mG1(u1)=1.

3. For vu2E(C3), we have mG1(v)=|E(H)|+2 and mG1(u2)=1.

4. For u1u2E(C3), we have mG1(u1)=mG1(u2).

By the construction of G2, we have the following:

1. For uvE(C4), we have mG2(u)=1 and mG1(v)=|E(H)|+1.

2. For vu1E(C4), we have mG2(v)=|E(H)|+1 and mG2(u1)=1.

3. For u1u2E(C4), we have mG2(u1)=|E(H)|+1 and mG2(u2)=1.

4. For u2uE(C4), we have mG2(u2)=1 and mG2(u)=|E(H)|+1.

Using the definition of the edge Mostar invariant and substituting the values from cases, we get the following:

Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q2|mG1(vi)mG1(vi+1)|+|mG1(v0)mG1(vq1)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1|mG2(v)mG2(vi)||mG2(v0)mG2(v1)||mG2(v1)mG2(v2)||mG2(v2)mG2(v3)||mG2(v3)mG2(v0)|,
Moe(G1)Moe(G2)=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21||E(H)|+q12(q12)|+i=q2q2|q12(|E(H)|+q12)|+||E(H)|+q12(q12)|u1u2E(H)|mG2(u1)mG2(u2)|i=4q1||E(H)|+q10)|4||E(H)|+q31|=u1u2E(H)|mG1(u1)mG1(u2)|+i=0q21|E(H)|+i=q2q2|E(H)|+|E(H)|u1u2E(H)|mG1(u1)mG1(u2)|i=4q1||E(H)|+q1|4||E(H)|+q4|q|E(H)|(q4)|E(H)|(q4)q+(q4)4|E(H)|4q+16q2+q+12<0.

This completes the proof. ∎

Theorem 2.1: Among all the cacti graphs in(n,s), the cactus^(n,s), forn3s+2,s2and forn9,s=1, shown inFigure 1has the largest edge Mostar invariant. Thus, for any cactusG(n,s), we haveMoe(G)Moe(˜(n,s)).

Proof: Let G(n,s) be a cactus graph where s0 and n2. If G˜(n,s) and G has a cut edge, then repeatedly applying Lemma 2.1, we get a sequence of new cacti graphs G1,G2,,Gb, where Gb is a cactus without any cut edge, such that Moe(G)<Moe(G1)<Moe(G2)<<Moe(Gb). Now, if Gb˜(n,s) and Gb have a cyclic subgraph G' that is constructed by attaching ri, for ri0, number of pendent vertices, ti, for ti0, number of C4 cycles and mi, for mi0, number of C3 cycles, at the vertices vi, for 1is1, of Cs, where s3, then by applying Lemma 2.2 repeatedly, we acquire a sequence of cacti graphs Gb,Gb1,Gb2,,Gbk satisfying Moe(Gb)<Moe(Gb1)<Moe(Gb2)<<Moe(Gbk), where Gbk is a cactus graph such that every vertex of cycles of Gbk has degree 2 except common vertices. If Gbk˜(n,s) and Gbk have a cycle Cq, for q5, then by applying Lemma 2.3 repeatedly, we acquire a sequence of cacti graphs Gbk,Gbk1,Gbk2,,Gbkc satisfying Moe(Gbk)<Moe(Gbk1)<Moe(Gbk2)<<Moe(Gbkc), where Gbkc˜(n,s). If Gbkc has a triangle C3 and at least one pendent edge vw, then by using Lemma 2.4, we construct a cactus graph Gbkc with a cycle C4 and get the greatest Mostar invariant and then Moe(Gbkc). This completes the proof. ∎ By Theorem 2.1 and simple calculation, we have the following results:

Corollary 2.1. LetG(n,s)be a cactus graph withn2and number of cycles s; then we have the following:

Moe(G){n23n+2,ifs=0andn2,n2n12,ifs=1andn9,n2+(2s3)n+s215s+2,ifs2andn3s+2,

equality holds if G˜(n,s).

3 Conclusion

The ongoing direction of numerical coding of the fundamental chemical structures with topological descriptors has been substantiated as completely victorious. This approach substantiates the contrast, quarry, renewal, interpretation, and swift troupe of chemical structures within enormous particularities. Eventually, topological descriptors can lead to productive measures for quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), which are imitations that identify chemical structures with chemical reactivity, physical properties, or biological activity. The edge Mostar index is a newly proposed quantity; it has not been used in physicochemical or biological research. Recently, a work (Imran et al., 2020) has been completed in this direction for chemical structures and nanostructures using graph operations. The authors have found the edge Mostar indices of nanostructures. Motivated by these results, we have studied the maximum edge Mostar invariant of the n-vertex cacti graphs with a fixed number of cycles in this study. For this, we have proved some lemmas in which we use the transformation of graphs and some calculations. In future, we want to find the largest and smallest edge Mostar invariants of the n-vertex cacti graphs with some fixed parameters other than the number of cycles.

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 author.

Author Contributions

FY: Data curation; investigation; methodology; project administration; software; validation. SA: Conceptualization; formal analysis; methodology; visualization. KA: Methodology; resources; visualization; writing-review and editing. SR: Visualization.

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.

The reviewer JL declared a past co authorship with the authors KA and SR to the handling editor.

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Keywords: topological invariants, Mostar invariant, edge Mostar invariant, cacti graphs, graph theory

Citation: Yasmeen F, Akhter S, Ali K and Rizvi STR (2021) Edge Mostar Indices of Cacti Graph With Fixed Cycles. Front. Chem. 9:693885. doi: 10.3389/fchem.2021.693885

Received: 12 April 2021; Accepted: 31 May 2021;
Published: 09 July 2021.

Edited by:

Jafar Soleymani, Tabriz University of Medical Sciences, Iran

Reviewed by:

Jia-Bao Liu, Anhui Jianzhu University, China
Micheal Arockiaraj, Loyola College, India

Copyright © 2021 Yasmeen, Akhter, Ali and Rizvi. 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: Farhana Yasmeen, farhanayasmeen.eu@gmail.com

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