# Sharp Lower Bounds of the Sum-Connectivity Index of Unicyclic Graphs

Authors: Maryam Atapour
Journal: Journal of Mathematics
 (2021)

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## Abstract

The sum-connectivity index of a graphG is defined as the sum of weights 1/du+dv over all edges uv of G, where du and dv are the degrees of the vertices u and v in graph G, respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n≥7 and diameter DG≥5.


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## Body

## 1. Introduction and Preliminaries

LetG be a simple graph with a vertex set V=VG and edge set EG. The integers n=nG=VG and m=mG=EG are the order and the size of the graph G, respectively. The open neighborhood of vertex v is NGv=Nv=u∈VG|uv∈EG, and the degree of v is dGv=dv=Nv. A pendant vertex is a vertex of degree one. The distance between two vertices is the number of edges in the shortest path connecting them, and the diameter DG of G is the distance between any two furthest vertices in G. A diametral path is the shortest path in G connecting two vertices whose distance is DG. A unicyclic graph is a connected graph containing exactly one cycle. A subgraph G′ of a graph G is a graph whose set of vertices is a subset of VG, and set of edges is a subset of EG.A topological index is a numeric number associated with a molecular graph that correlates certain physicochemical properties of chemical compounds. The topological indices are useful in the prediction of physicochemical properties and the bioactivity of the chemical compounds [1–3]. Also, topological indices invariants are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) studies. It was demonstrated that the sum-connectivity index is well correlated with a variety of physicochemical properties of alkanes, such as boiling point and enthalpy of formation. The sum-connectivity index is certainly the most widely applied in chemistry and pharmacology, in particular for designing quantitative structure-property and structure-activity relations. The sum-connectivity index is proposed to quantitatively characterize the degree of molecular branching.Topological indices have been used and have been shown to give a high degree of predictability of pharmaceutical properties. The sum-connectivity index of a graphG was proposed in [4] defined as follows:(1)SCIG=∑uv∈EG1du+dv.The applications of the sum-connectivity index have been investigated in [5, 6]. Some basic mathematical properties of the sum-connectivity index have been established in [4–8].In [4], it was shown that for a graph G with n≥5 vertices and without isolated vertices, SCIG≥n−1/n with equality if and only if G is the star. For n=4, this is not true since, for the union of two copies of the path on two vertices, its sum-connectivity index is 2, less than 3/2. In [7], minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number are characterized; in [8], sum-connectivity index of molecular trees are characterized; and in [4], some of the lower and upper bounds for the sum-connectivity index of trees are obtained (see recent bounds [9–13]). We all know that the sum-connectivity index is one of the most important and practical indices and therefore has been considered by many researchers. In this paper, we will address one of the unresolved issues for the sum-connectivity index. In fact, we investigate the relationship between the sum-connectivity index and diameter of the graph, which is one of the important parameters in graph theory and we get new results. In other words, in this paper, we solve the problem of the relationship between the diameter of a graph and the sum-connectivity index for the unicyclic graph.


## 2. Main Results

We begin with the following lemma that we will need for obtaining our main results.We denotetG as the number of pendant vertices in a graph G.Lemma 1.
LetG be any unicyclic graph and U be a diametral path of G. If G contains a pendant vertex v not in U, then there is a unicyclic graph G′⊂G not containing v, such that DG=DG′,tG′=tG−1, and SCIG′<SCIG.Proof.
LetU be a diametral path of G and v∈VG be a pendant vertex not in U. We denote by u the closest vertex to v which is not of degree 2. Let G′ be a subgraph of G obtained by the removal of the path connecting u and v from G. Let u′ be the neighbor of u on the u−v path (if the path has only one edge, then u′=v). Clearly, G′ is a unicyclic graph, DG′=DG, and tG′=tG−1. Furthermore,(2)SCIG−SCIG′≥1du+1+∑w∈Nu\u′1du+dw−1du+dw−1.
We know that the sum-connectivity index of the pathuv is at least 1/du+1. Note that(3)1du+dw−1du+dw−1≥1du+1−1du,for every w∈Nuu′; hence, we have(4)SCIG−SCIG′≥1du+1+du−11du+1−1du.
Therefore, we getSCIG>SCIG′.By Lemma1, it follows that for any unicyclic graph G, if U is a diametral path of G, then there is a unicyclic graph G′⊂G containing only pendant vertices of U, where DG′=DG and SCIG′<SCIG.Here, we obtain a sharp bound on the sum-connectivity index of any unicyclic graph of diameter at least 5.Theorem 2.
LetG be any unicyclic graph of diameter DG≥5. Then,(5)SCIG≥DG2+55+13−2.Proof.
We will complete the proof by considering the following four cases.Case 1. IfG does not contain any pendant vertex, then G is the cycle either with 2DG or 2DG+1 vertices, which implies that(6)SCIG≥2DG14=DG>DG2+55+13−2.Case 2. IfG contains one pendant vertex, then G consists of the cycle ℂr of length r≥3 and the path P having s≥1 edges, where ℂr∩P consists of one vertex which has degree 3 in G. This degree will be included in the computation of SCIℂr and SCIP. We have SCIG=SCIℂr+SCIP, where(7)SCIℂr=∑uv∈Eℂr1du+dv=r−214+215=r2+255−1,SCIP=∑uv∈EP1du+dv=s−214+13+15=s2+33+55−1.Ifs≥2 and SCIP=1/2 if s=1. So, SCIP≥s/2 for every s≥1, and the equality holds if s=1.Ifr≥4, then DG≤r/2+s and(8)SCIG≥r2+255+s2−1≥r2+255+DG2−r4−1=DG2+r4+255−1≥DG2+255>DG2+55+13−2.Ifr=3, then DG=s+1 and(9)SCIG≥32+255+s2−1,≥32+255+DG2−12−1=DG2+255>DG2+55+13−2.Case 3. If G contains 2 pendant vertices, then G consists of the cycle ℂr of length r≥3 and two paths X1 and X2 having s1≥1 and s2≥1 edges, respectively. We can assume that ℂr∩X1 consists of one vertex and X2 is attached either to an internal vertex of X1 or to a vertex of ℂr.Case 3.1. If X1∩X2=∅, then we have SCIG=SCIℂr+SCIX1+SCIX2. For i=1,2,(10)SCIXi=∑uv∈EXi1du+dv=si−214+13+15=si2+33+55−1.Whensi≥2 and SCIXi=1/2 if si=1. If X1 and X2 are attached to nonadjacent vertices of ℂr, then(11)SCIℂr=∑uv∈Eℂr1du+dv=r−414+415=r2+455−2.IfX1 and X2 are attached to adjacent vertices of ℂr, then(12)SCIℂr=∑uv∈Eℂr1du+dv,=r−314+215+16=r2+255+66−32>r2+455−2.Ifs1=s2=1, then DG≤r/2+2 and(13)SCIG≥r2+455−2+12+12,≥DG+455−3>DG2+55+13−2.So, we can assume thats1 or s2 is at least 2. We have SCIX1+SCIX2≥s1/2+s2/2+3/3+5/5−1 (the equality holds if s1 or s2 is 1).Ifr≥4, then DG≤r/2+s1+s2 and(14)SCIG≥r2+455−2+s12+s22+33+55−1,≥DG2+r4+5+33−3≥DG2+5+33−2.Ifr=3, then DG≤s1+s2+1 and(15)SCIG≥32+455−2+s12+s22+33+55−1,≥DG2+455+33+55−2=DG2+5+33−2.Case 3.2.X1∩X2 is nonempty, and there is a diametral path containing both pendant vertices of G.LetU be the diametral path containing both pendant vertices of G. Then, U⊆X1∪X2. One of the internal vertices, say x, of U is of degree 3 or 4 in graph G. If x is adjacent to a pendant vertex of U, then(16)SCIU=∑uv∈EU1du+dv=DG−314+13+15+16=DG2+33+55+66−32.Ifx is not adjacent to a pendant vertex of U, then(17)SCIU≥DG−42+233+266=DG2+233+266−2.Note thatG contains also the cycle ℂr, where one of the vertices is of degree 3 or 4 in G. We have(18)SCIℂr=∑uv∈Eℂr1du+dv=r−214+216≥32+266−1=12+266,which implies that(19)SCIG≥SCIU+SCIℂr≥DG2+33+55+66−32+12+266,=DG2+33+55+62−1>DG2+55+13−2.Case 3.3.X1∩X2 is nonempty and there is a diametral path containing only one pendant vertex of G. Then, we denote this diametral path with U containing only one pendant vertex of G. Since the other pendant vertex is not in U, by Lemma 1, there is a unicyclic graph G′ having one pendant vertex, such that DG′=DG and SCIG>SCIG′, and we know that SCIG′>DG/2+5/5+1/3−2.Case 4. IfG contains at least 3 pendant vertices.
LetU be a diametral path of G. Clearly, this path contains at most 2 pendant vertices of G. Since G contains m≥3 pendant vertices, we have at least m−2 pendant vertices not in U. By Lemma 1, there is a unicyclic graph G′⊆G having only the pendant vertices of U (at most 2 vertices), such that DG′=DG,tG′=tG−1 and SCIG>SCIG′. From the previous cases, it follows that SCIG′>DG/2+5/5+1/3−2.
It is easy to show that the boundSCIG≥DG/2+5/5+1/3−2 is best possible because of the graph H, where VH=u,v0,v1,v2,…,vDH and EH=v0v1,v1v2,…,vDH−1vDH,uv1,uv3 has the sum-connectivity index(20)SCIH=DH−414+515+13,=DH2+55+13−2.
The proof is completed.Now, we obtain lower bounds on the sum-connectivity index for unicyclic graphs of small diameter.Theorem 3.
LetG be an unicyclic graph of diameter DG. Then,(i)
IfDG=2, then SCIG≥1+25/5(ii)
IfDG=3, then SCIG≥25/5+6/6+1(iii)
IfDG=4, then SCIG≥1+45/5Proof.
We can see that the proof of Theorem2 holds for DG=3 and DG=4 except for Case 3.1, where s1=s2=1.
LetDG=4. We have SCIG≥DG+1+45/5−3 (as presented in the proof of Theorem 2), which is 1+45/5. From the other cases, we obtain SCIG≥DG/2+5/5+1/3−2=5+1/3>1+45/5, which implies that SCIG≥1+45/5.
LetDG=3. We have SCIG≥DG+45/5−3 (obtained in the proof of Theorem 2); Case 3.1 (if s1=s2=1) is not sufficient now, so we give a better bound in this case. Since s1=s2=1 and DG=3, then P1 and P2 must be attached to adjacent vertices of Cr, which means that SCICr=r/2+25/5+6/6−3/2 in the proof of Theorem 2. Since r≥3, we obtain(21)SCIG=SCICr+SCIX1+SCIX2=r2+255+66−32+12+12≥255+66+1.
LetDG=2. Except for ℂ5 and ℂ4, the only unicyclic graphs H of diameter 2 are formed by the cycle ℂ3, where s≥1 pendant vertices are adjacent to one of the vertices of C3. Let Vℂ3=v1,v2,v3. We can assume that the pendant vertices u1,u2,…,up are adjacent to v1. Then, U=v2v1u1 is a diametral path of H, and from Lemma 1, it follows that there is a unicyclic graph H′⊆H, which contains only one pendant vertex u1 (the pendant vertex (the pendant vertex in U)), where SCIH≥SCIH′. Since(22)SCIH′=215+214=1+255,
SCIℂ4=2 and SCIℂ5=5/2, we obtain the bound SCIG≥1+25/5.Corollary 4.
LetG be any unicyclic graph of order at least 7 and diameter DG≥2. Then,(23)SCIG≥DG2+55+13−2.Proof.
By Theorem2, for DG≥5 and any n, we have SCIG≥DG/2+5/5+1/3−2. By Theorem 3, for DG=2 and any n, we have SCIG≥1+25/5, which is greater than DG/2+5/5+1/3−2. It remains to prove Corollary 4 for n≥7 and 3≤DG≤4. The proof of Theorem 2 holds also for DG=3 and DG=4 except for Case 3.1 where s1=s2=1. We show that if n≥7, then SCIG≥DG/2+5/5+1/3−2 also in that case. If n≥7 and p1=p2=1, then G contains the cycle Ck for r≥5 and SCIℂr≥r/2+45/5−2 (given in Case 3.1 in the proof of Theorem 2). Since SCIX1=SCIX2=1/2, we obtain(24)SCIG=SCICr+SCIX1+SCIX2≥32+455,which is greater than SCIG≥DG/2+5/5+1/3−2 for DG=3 and DG=4.Corollary 5.
LetG be any unicyclic graph of order at least 7 and diameter DG≥2. Then,(25)SCIGDG≥3/3+5n−2−12,SCIG−DG≥33+5−n2−1.Proof.
By Theorem2, we have SCIG≥DG/2+5/5+1/3−2 and since DG≤n−2 for any graph G except for the path, hence, by the definition of sum-connectivity index, we have(26)SCIGDG≥12+55DG+13DG−2DG,≥12+55n−2+13n−2−2DG≥12+55n−2+13n−2−1=3/3+5n−2−12.
Similarly, we obtain(27)SCIG−DG≥33+5−DG2−2≥33+5−n2−1.


## 3. Open Problem and Conclusion

In this paper, we investigate the relationship between the sum-connectivity index and the diameter of a graph and obtained a new lower bound for the sum-connectivity index of unicyclic graphs. However, there are still open and challenging problems for researchers, for example, the problem on the relationship between the sum-connectivity index and the diameter of bicyclic and tricyclic graphs. Moreover, the relationship between other topological indices such as F-index and GA-index with the diameter of unicyclic, bicyclic, and tricyclic graphs is still open.



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