# Leap Eccentric Connectivity Index of Subdivision Graphs

Authors: Ali Ghalavand; Shiladhar Pawar; Nandappa D. Soner
Journal: Journal of Mathematics
 (2022)

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## Abstract

The second degree of a vertex in a simple graph is defined as the number of its second neighbors. The leap eccentric connectivity index of a graphM, LξcM, is the sum of the product of the second degree and the eccentricity of every vertex in M. In this paper, some lower and upper bounds of LξcSM in terms of the numbers of vertices and edges, diameter, and the first Zagreb and third leap Zagreb indices are obtained. Also, the exact values of LξcSM for some well-known graphs are computed.


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## Body

## 1. Introduction

In this paper,M is a finite and undirected simple graph. Let VM and EM be sets of vertices and edges of M, respectively. Then, we put n=VM and m=EG. If a,b⊆VM, then the length of a shortest path connecting a and b in M is the distance between a and b in M and denoted by dMa,b. Let x be a vertex of M, and let r be a positive integer. Then, the open r-neighborhood of x in M, Nrx, is the set of all vertices at distance r from x; that is, Nrx=v∈VM:dMv,x=r. The r-distance degree of a vertex x in M is the size of the open r-neighborhood of x in M, and it is denoted by drx/M or simply drx if no misunderstanding is possible; that is, drx/M=drx=Nrx. It is clear that d1x/M is the degree of vertex x in M, and we denoted it by dMx or simply dx. Also, the eccentricity of a vertex x in M, ev, is defined as ev=maxdMv,u:u∈VM, and the diameter and radius of graph M are defined as diamM=maxev:v∈VM and radM=minev:v∈VM, respectively.The subdivision graphSM of a simple graph M is the graph obtained from M by inserting an additional vertex into each edge of M, or equivalently, by replacing each of its edges with a path of length 2 [1].The wheel graphW1,q of order q+1 is the join of K1 and Cq in which K1 is the complete graph with one vertex, and Cq is the q -vertex cycle graph. Clearly, VW1,q=q+1 and EW1,q=2q. The apex vertex of the wheel is the vertex corresponding to K1, and the rim vertices of the wheel are the vertices corresponding to Cq [2]. Note that all notions and notations not defined here can be obtained from the book of Harary [2].In chemical graph theory, a numerical parameter of a given graph that is applicable in some chemical problems is called a topological index. The Zagreb group indices are two degree-based topological indices that were defined by Gutman and Trinajestic [3] in 1972 and elaborated in [4]. These indices are defined as(1)M1M=∑x∈VMdMx2,M2M=∑ab∈EMdMadMb.For the main properties of these two indices, we refer the interested readers to [3–7].In 2017, Naji et al. [8] introduced three topological indices depending on the second degree of vertices. These invariants are so-called leap Zagreb topological indices and can be defined as follows:(2)LM1M=∑v∈VMd2v2,LM2M=∑uv∈EMd2ud2v,LM3M=∑v∈VMdud2v.In [9], the first leap Zagreb topological index of some graph operations is computed, and in [10], some formulas for the leap Zagreb indices of generalized rts point line transformation graphs TrtsM, when s=1, are obtained. We refer to [8–14] for more details on the leap Zagreb indices of graphs. In [15], Sharma et al. introduced the eccentric connectivity index of the graph M as ξcM=∑v∈VMdvev. For mathematical properties, the interested readers can consult [15–17].Recently, authors found in [18] introduced the leap eccentric connectivity index of a graph M. It is denoted by LξcM and can be defined LξcM=∑v∈VMd2vev. They obtained the exact values of the leap eccentric connectivity index of complete, complete bipartite, cycle, path, and wheel graphs and determined some upper and lower bounds for LξcM in terms of the number of vertices, number of edges, diameter, total eccentricity, and Zagreb indices. In [19], the explicit formulas of the leap eccentric connectivity index for the Cartesian product, composition, disjunctions, symmetric difference, and corona product were computed.In [20], exact values of Lξc for thorny complete graphs, thorny complete bipartite graphs, thorny cycles, and thorny paths were reported. The authors of this paper also discussed some applications of the leap eccentric connectivity index of chemical structures such as cyclo-alkanes. In [21], some new upper and lower bounds for LξcM in the terms of the order, size, diameter, radius, and total eccentricity, Zagreb, and leap Zagreb indices are found. In the mentioned paper, some lower and upper bounds of LξcSM in terms of the numbers of vertices and edges, diameter, and the first Zagreb and third leap Zagreb indices are also obtained. They also found the exact values of LξcSM for some well-known graphs.The following results of [18,22] are crucial in our arguments:Theorem 1 (see [18]).
Letn≥3 be an integer. Then,(3)LξcPn=3n2−10n+122,2|n,3n2−10n+112,2|n.Theorem 2 (see [18]).
Letn≥3 be an integer. Then,(4)LξcCn=0,n=3,8,n=4,n2,n≠4,2∣n,n(n−1),n≠3,2∤n.Lemma 1 (see [22]).
LetM be an n-vertex connected graph of size m. Then,(5)d2v≤∑u∈N1vd1u−d1v.The equality is attained if and only if G is a C3,C4-free graph.By Lemma1, for a C3,C4-free graph M, we have ∑v∈VGd2v=M1−2m.


## 2. Main Results

The aim of this paper is to present the exact values of leap eccentric connectivity index of subdivision graph of some standard graphs.Theorem 3.
Supposen≥3. Then,(6)LξcSKn=36,ifn=3,nn−14n−5,otherwise.Proof.
Leta1,a2,…,an be the vertices of Kn, and let b1,b2,…,bm be the new vertices added to Kn to obtain SKn, where m is the size of Kn. Then, d2ai=n−1, d2bj=2n−4, eai=3, and ebj=3,ifn=3,4,otherwise.
By definition, we have two following cases:Case 1.
Ifn=3, then(7)LξcSK3=∑i=1623=66=36.Case 2.
Ifn≥4, then(8)LξcSKn=∑w∈VSKnd2wew=∑i=1nd2aieai+∑i=1md2bjebj=∑i=1nn−13+∑j=1m2n−44=3nn−1+8mn−2.
Since for the complete graphKn, m=nn−1/2, it follows that
LξcSKn=nn−14n−5.Theorem 4.
Forr≥s≥2, let Kr,s be the complete bipartite graph. Then,(9)LξcSKr,s=rs3r+3s+2.Proof.
Supposer≥s≥2 and V1,V2 is a partition of the vertex set, where V1=v1,v2,v3,…,vr,V2=u1,u2,u3,…,us and let W=w1,w2,w3,…,wrs be the set of new vertices in SKr,s. Then, d2vi=s, d2uj=r, d2wk=r+s−2, evi=4, euj=4, and ewk=3. By definition,(10)LξcSKr,s=∑vi∈V1d2v.ev+∑uj∈V2d2u.eu+∑wk∈V3d2w.ew=∑i=1rs4+∑j=1sr4+∑k=1rsr+s−23=4rs+4rs+3rsr+s−2=rs3r+3s+2.Theorem 5.
LetK1,n−1 be the star graph of order n≥3. Then,(11)LξcSK1,n−1=3nn−1.Proof.
Letv0∈K1,n−1, with dv0=n−1, are be the central vertex, v1,v2,…,vn−1 are be the pendent vertices of K1,n−1, and u1,u2,…,un−1 are be the new vertices added to K1,n−1, to obtain SK1,n−1. If i=1,2,…,n−1, then d2v0=n−1, d2vi=1, d2ui=n−2, ev0=2, evi=4, and eui=3. By definition,(12)LξcSK1,n−1=d2v0ev0+∑i=1n−1d2vievi+∑j=1n−1d2ujeuj=n−12+∑i=1n−114+∑j=1n−1n−23=3nn−1.Theorem 6.
Letr≥1 and s≥1 be two integers such that n=r+s≥3. Then,(13)3nn−1≤LξcSKr,s≤14n23n+2.On the left hand side, equality occurs if and only ifKr,s≅K1,n−1. On the right hand side, equality occurs if and only if Kr,s≅Kn/2,n/2.Proof.
We consider two cases as follows:(i)
r=1 or s=1. In this case, G≅K1,n−1, and by Theorem 5, LξcSKr,s=3nn−1.(ii)
s,r≥2. In this case, since r+s=n, 2n−2≤rs≤n/2n/2. Now, by Theorem 4, 2n−23n+2≤LξcSKr,s≤1/4n23n+2. On the left hand side, equality holds if and only if Kr,s≅K2,n−2. On the right hand side, equality holds if and only if Kr,s≅Kn/2,n/2.
On the other hand,2n−23n+2−3nn−1=n3n−5−8>0. Therefore, by (i) and (ii), 3nn−1≤LξcSKr,s≤1/4n23n+2. On the left hand side, equality occurs if and only if Kr,s≅K1,n−1, and on the right hand side, equality occurs if and only if Kr,s≅Kn/2,n/2.Proposition 1.
Letn be an integer. Then,(14)LξcSCn=4n2.Proof.
SinceSCn=C2n, the proof follows from Theorem 2.Proposition 2.
Letn≥2 be an integer. Then, LξcSPn=23n2−8n+6.Proof.
SinceSPn=P2n−1, the proof follows from Theorem 1.Theorem 7.
Forn≥6, LξcSW1,n=2n2n+23.Proof.
Letv0 be the central vertex of W1,n, v1,v2,…,vn, be the rim vertices W1,n and let SW1,n be the subdivision of W1,n. If wi subdivides v0vi, 1≤i≤n, uj subdivides vjvj+1, 1≤j≤n−1 and un subdivides vnv1. One can easily verify LξcSW1,n=120, if n=3, LξcSW1,n=204, if n=4 and LξcSW1,n=310, if n=5. Let n≥6. Then, d2v0=n, d2vi=3, d2uj=4, d2wk=n+1, ev0=3, evi=5, euj=6, ewk=4, 1≤i,j,k≤n. By definition, we have(15)LξcSW1,n=∑v∈VSW1,nd2vev=d2v0ev0+∑i=1nd2vievi+∑i=1nd2ujeuj+∑i=1nd2wkewk=n3+∑i=1n35+∑i=1n46+∑i=1nn+14=2n2n+23.Theorem 8.
For natural numbersr and s, let Dr,s be a double star with v1,v2,v3,…,vr be the pendent vertices have support at v0 and u1,u2,u3,…,us, be the pendent vertices have support at u0. Then,(16)LξcSDr,s=5r2+s2+13r+s+8.Proof.
Letxi subdivides v0vi, 1≤i≤r, yj subdivides u0ui, 1≤j≤s, and w0 subdivides v0u0. Then, d2v0=r+1 , d2u0=s+1, d2w0=r+s, d2vi=1, d2uj=1, d2xi=r, d2yj=s, ev0=4, eu0=4, ew0=3evi=6, euj=6, exi=5, and eyj=5. By definition, we have(17)LξcSDr,s=d2v0ev0+∑i=1rd2xiexi+∑i=1rd2vievi+d2w0ew0+d2u0eu0+∑j=1sd2yjeyj+∑j=1sd2ujeuj=r+14+∑i=1rr5+∑i=1r16+r+s3+s+14+∑j=1ss5+∑j=1s16=5r2+s2+13r+s+8.Theorem 9.
Letn≥7 be a natural number. Then,(18)LξcSDi,n−2−i>LξcSDi+1,n−3−ifori=1,2,…⌊n−22⌋−1.Proof.
By Theorem8,(19)LξcSDi,n−2−i−LξcSDi+1,n−3−i=10n−2i−3.
Now, if2|n−2, then by (19),(20)LξcSDi,n−2−i−LξcSDi+1,n−3−i≥10n−2n−22−1−3=10.
And if2∤n−2, then by (19),(21)LξcSDi,n−2−i−LξcSDi+1,n−3−i≥10n−2n−32−1−3=20.Therefore, LξcSDi,n−2−i>LξcSDi+1,n−3−i for i=1,2,…⌊n−2/2⌋−1.Corollary 1.
Letr, s, and n be three natural numbers such that r+s+2=n≥7. Then,(22)52n2+3n−8≤LξcSDr,s≤5n2−17n+32,2|n−2,125n+1n−3≤LξcSDr,s≤5n2−17n+32,2|n−2.
On the left hand side, equalities occur if and only ifDr,s≅D⌊n−2/2⌋,⌊n−2/2⌋. On the right hand side, equalities occur if and only if Dr,s≅D1,n−3.Theorem 10.
LetM be an n-vertex connected graph of size m such that n≥3. Then,(23)LξcM≤nM1M−2nm−LM3M.The bound is attained forP4.Proof.
Sinceev≤n−dv for every v∈VM,(24)LξcM=∑v∈VMd2vev≤∑v∈VMd2vn−dv=∑v∈VMnd2v−∑d1vd2v=n∑v∈VMd2v−∑d1vd2v.
Using definition ofLM3M and Lemma 3, we get(25)LξcM≤n∑v∈VM∑uv∈EMdu−dv−LM3M=n∑v∈VMdv2−2nm−LM3M=nM1M−2nm−LM3M.Corollary 2.
LetM be an n-vertex connected graph of size m such that n≥3. Then,(26)LξcSM≤n+m−3M1M+4m.Proof.
Foruv∈EM, let vuv be the new vertex of degree 2 on uv in SM. By definition of SM, dv/SM=dv/M, d2v/SM=d2v/M for v∈VM and dvuv/SM=2, d2vuv/SM=du/M+dv/M−2 for uv∈EM. Therefore, M1SM=M1M+4m and LM3SM=M1M+2M1M−4m=3M1M−4m. So, by Theorem 10, LξcM≤n+m−3M1M+4m.Theorem 11.
LetM be an n-vertex connected graph of size m≥2. Then, LξcSG≥2n+m.Proof.
LetV0=v∈VM;dv=n−1 and n0=V0. Then, d2v=0 for every v∈V0 and for every u∈V\V0, we have eu≥2 and d2u≥1. Hence,(27)LξcM=∑v∈V0d2vev+∑v∈V/V0d2vev≥∑v∈V001+∑v∈V/V012=2V/V0=2n−n0.
Now, it is easy to see that the number of vertices ofSM is n+m, and the number of vertices of degree n−1 in SM is zero. Therefore, by (27), we have LξcSM≥2nSM−n0SM=2n+m.Theorem 12.
LetM be an n-vertex graph of size m. Then,(28)LξcM≤diamMM1M+2m.The equality occurs if and only ifM is a self-centered and C3,C4-free graph.Proof.
By definition, for allv∈VM, ev≤diamM, the equality holds if and only if M is a self-centered. Also, by Lemma 3, ∑v∈VMd2v≤M1M−2m, and the equality occurs if and only if M is a C3,C4-free graph. Therefore,(29)LξcM=∑v∈VMd2vev≤∑v∈VMd2vdiamM≤diamMM1M−2m.
The equalities hold if and only ifM is a self-centered and C3,C4-free graph.Corollary 3.
LetM be an n-vertex graph of size m. Then,(30)LξcSM≤diamSMM1M.
The equality occurs if and only ifSM is a self-centered.Theorem 13.
LetM be an n-vertex connected graph of size m such that n≥4. Then, LξcSM≥4M1M−2m, the equality occurs if and only if M≅Kn.Proof.
By definition ofSM, for all v∈VM, d2v/SM=dv/M, ev≥3, and the equalities occur if and only if M≅Kn. Also, for all uv∈EM, d2vuv/SM=dv/M+du/M−2, evuv≥4, and the equalities occur if and only if M≅Kn. Therefore, by definitions of Lξc and SM, we have(31)LξcSM=∑v∈VMd2vSMevSM+∑uv∈EMd2vuvSMevuvSM≥∑v∈VM3dvM+∑uv∈EM4duM+dvM−2=6m+4M1M−8m=4M1M−2m.
The equality occurs if and only ifM≅Kn.



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