# Modified Eccentric Connectivity of Generalized Thorn Graphs

Authors: Nilanjan De; Anita Pal; Sk. Md. Abu Nayeem
Journal: International Journal of Computational Mathematics
 (2014)

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## Abstract

The thorn graphGT of a given graph G is obtained by attaching t(>0) pendent vertices to each vertex of G. The pendent edges, called thorns of GT, can be treated as P2 or K2, so that a thorn graph is generalized by replacing P2 by Pm and K2 by Kp and the respective generalizations are denoted by GPm and GKp. The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph GT and the generalized thorn graphs GKp and GPm.


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## Body

## 1. Introduction

LetG be a simple connected graph with vertex set V(G) and edge set E(G), so that |V(G)|=k and |E(G)|=e. Let the vertices of G be labeled as v1,v2,…,vk. For any vertex vi∈V(G) the number of neighbors of vi is defined as the degree of the vertex vi and is denoted by dG(vi). Let N(vi) denote the set of vertices which are the neighbors of the vertex vi, so that |N(vi)|=dG(vi). Also let δG(vi)=∑vj∈N(vi)dG(vj), that is, sum of degrees of the neighboring vertices of vi∈G. The distance between the vertices vi and vj is equal to the length of the shortest path connecting vi and vj. Also for a given vertex vi∈V(G), the eccentricity εG(vi) is the largest distance from vi to any other vertices of G and the sum of eccentricities of all the vertices of G is denoted by θ(G) [1]. The eccentric connectivity index of a graph G was proposed by Sharma et al. [2]. A lot of results related to chemical and mathematical study on eccentric connectivity index have taken place in the literature [3–5]. There are numerous modifications of eccentric connectivity index reported in the literature till date. These include edge versions of eccentric connectivity index [6], eccentric connectivity topochemical index [7], augmented eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and connective eccentricity index [10]. A modified version of eccentric connectivity index was proposed by Ashrafi and Ghorbani [11].Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as(1)ξcG,x=∑i=1k‍δG(vi)xεG(vi),
so that the modified eccentric connectivity index is the first derivative of this polynomial for x=1. Several studies on this modified eccentric connectivity index are also found in the literature. In [11], the modified eccentric connectivity polynomials for three infinite classes of fullerenes were computed. In [12], a numerical method for computing modified eccentric connectivity polynomial and modified eccentric connectivity index of one-pentagonal carbon nanocones was presented. In [13], some exact formulas for the modified eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction, and join of graphs were presented. Some upper and lower bounds for this modified eccentric connectivity index are recently obtained by the present authors [14].The first and the second Zagreb indices ofG, denoted by M1(G) and M2(G), respectively, are two of the oldest topological indices introduced in [15] by Gutman and Trinajstić and were defined as
(2)M1G=∑i=1k‍dGvi2=∑(vi,vj)∈EGdGvi+dGvj=∑i=1k‍δGvi,M2G=∑(vi,vj)∈E(G)‍dG(vi)dG(vj).Letp=(p1,p2,…,pk) be a k-tuple of nonnegative integers. The thorn graph GT is the graph obtained from G by attaching pi pendent vertices to the vertex vi,i=1,2,…,k of G. In this paper, we assume p1=p2=⋯=pk=t. These pendent vertices are termed as thorns. The concept of thorn graphs was first introduced by Gutman [16]. A lot of studies on thorn graphs for different topological indices are made by several researchers in the recent past [17–24]. Very recently, De [25, 26] studied two eccentricity related topological indices, such as eccentric connectivity index and augmented eccentric connectivity indices, on thorn graphs.The thorns of the thorn graphGT can be treated as P2 or K2, so that the thorn graph can be generalized by replacing P2 by Pm and K2 by Kp and the generalizations are, respectively, denoted by GPm and GKp. In the following section, we present the explicit expressions of the modified eccentric connectivity index of thorn graph GT and its generalized forms GKp and GPm.


## 2. Evaluation of Modified Eccentric Connectivity Index

The eccentric connectivity indexξc(G) [2] and connective eccentric index Cξ(G) [10] of a graph G are defined as
(3)ξc(G)=∑i=1k‍dG(vi)εG(vi),Cξ(G)=∑v∈V(G)‍dG(v)εG(v).The modified eccentric connectivity indexξc(G) [11] is defined as
(4)ξc(G)=∑i=1k‍δG(vi)εG(vi).Total eccentricity index is defined asζ(G)=∑v∈V(G)‍ϵG(v). Total eccentricity index of the generalized hierarchical product of graphs has been studied by De et al. recently [14].Since the modified eccentric connectivity index is likely to have an application in drug discovery process, therefore, we evaluate the index in comparison to some other well known indices in this section.The two graphs shown in Figure1 have the same eccentricity connectivity index and connective eccentricity index, but they have different modified eccentric connectivity indices.Figure 1
Two graphs with the sameξc and Cξ, but with different ξc.To evaluate the modified eccentric connectivity index (MECI) in terms of degeneracy and intercorrelation with other well known indices we compute different topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), connective eccentricity index, (CEI) and augmented eccentric connectivity index (AECI) for octane isomers as given in Table1.Table 1
Different topological indices of octane isomers.
Molecule
MECI
ECI
TEI
CEI
AECI
Octane
136
74
44
2.352
3.838
2-Methyl-heptane
126
65
39
3.167
6.167
3-Methyl-heptane
121
63
38
3.25
6.267
4-Methyl-heptane
118
61
37
3.383
6.55
3-Ethyl-hexane
107
54
33
3.767
7.867
2,2-Dimethyl-hexane
132
56
34
3.633
7.8
2,3-Dimethyl-hexane
117
54
33
3.767
7.6
2,4-Dimethyl-hexane
117
54
33
3.767
7.933
2,5-Dimethyl-hexane
114
56
34
3.633
7.4
3,3-Dimethyl-hexane
114
52
32
5.4
8.3
3,4-Dimethyl-hexane
112
52
32
3.9
7.8
2-Methyl-3-ethyl-pentane
93
43
27
4.833
11.5
3-Methyl-3-ethyl-pentane
95
41
26
5.083
10.833
2,2,3-Trimethyl-pentane
107
43
27
4.833
10.5
2,2,4-Trimethyl-pentane
115
45
28
4.583
11.833
2,3,3-Trimethyl-pentane
103
41
26
5.083
10.333
2,3,4-Trimethyl-pentane
101
43
43
4.833
11.5
2,2,3,3-Tetramethyl-butane
100
34
34
6
10For modified eccentric connectivity index we observe that maximum value = 136, minimum value = 93, ratio = 1.46, and degeneracy = 4/18.Intercorrelation of modified eccentric connectivity index with some well known vertex eccentricity based topological indices is given in Table2.Table 2
Intercorrelation of indices.
MECI
ECI
TEI
CEI
AECI
MECI
1
0.737
0.325
0.628
0.652
ECI
1
0.426
0.866
0.852
TEI
1
0.34
0.405
CEI
1
0.689
AECI
1


## 3. Main Results

First we find the modified eccentric connectivity index of the thorn graphGT in terms of modified eccentric connectivity index of G, total eccentricity of G, and first Zagreb index of G.Theorem 1.
For any simple connected graphG, ξc(GT) and ξc(G) are related as ξc(GT)=ξc(G)+2tξc(G)+t(t+1)θ(G)+M1(G)+6et+kt+2kt2, where |V(G)|=k and GT is the thorn graph of G.Proof.
LetV(G)={v1,v2,…,vk} and GT=(VT,ET), where VT=VG∪V1∪V2⋯∪Vk. Here, Vi are the set of degree one vertices attached to the vertices vi in GT and Vi∩Vj=φ,i≠j. Let the vertices of the set Vi be denoted by vi1,vi2,…,vik for i=1,2,…,k.
Then the degree of the verticesvi in GT is given by dGT(vi)=dG(vi)+t, for i=1,2,…,k. Hence, δGT(vi)=δG(vi)+dG(vi)t+t for i=1,2,…,k, and δGT(vij)=dG(vi)+t. Similarly the eccentricity of the vertices vi, i=1,2,…,k in GT is given by εGT(vi)=εG(vi)+1, for i=1,2,…,k, and the eccentricity of the vertices vij is given by εGT(vij)=εG(vi)+2, for j=1,2,…,t and i=1,2,…,k.
Then the modified eccentric connectivity index ofGT is given by
(5)ξc(GT)=∑i=1k‍δGT(vi)εGT(vi)+∑i=1k‍∑j=1t‍δGT(vij)εGT(vij).Now,
(6)∑i=1k‍δGTviεGTvi=∑i=1k‍δGvi+tdGvi+tεGvi+1=ξcG+tξcG+tθG+M1G+2et+kt,∑i=1k‍∑j=1t‍δGTvijεGTvij=∑i=1k‍∑j=1t‍dGvi+tεGvi+2=∑i=1k‍∑j=1t‍dGviεGvi+∑i=1k‍∑j=1t‍tεGvi+2∑i=1k‍∑j=1t‍dGvi+∑i=1k‍∑j=1t‍2t=tξc(G)+t2θ(G)+4et+2kt2.
Combining the above equations, we get(7)ξcGT=ξcG+2tξcG+tt+1θG+M1G+6et+kt+2kt2.The eccentric connectivity polynomial and total eccentricity polynomial ofG are defined as ξc(G,x)=∑i=1kdG(vi)xεG(vi) and θ(G,x)=∑i=1kxεG(vi), respectively. It is easy to see that the eccentric connectivity index and the total eccentricity of a graph can be obtained from the corresponding polynomials by evaluating their first derivatives at x=1.Now we express the modified eccentric polynomial of a thorn graph in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph.Theorem 2.
For any simple connected graphG, the polynomials ξc(GT,x) and ξc(G,x) are related as ξc(GT,x)=xξc(G,x)+tx(x+1)ξc(G,x)+tx(tx+1)θ(G,x), where GT is the thorn graph of G.Proof.
Following the previous theorem, the modified eccentric connectivity polynomial ofGT is given by ξc(GT,x)=∑i=1kδGT(vi)xεGT(vi)+∑i=1k∑j=1tδGT(vij)xεGT(vij).
Now,(8)∑i=1k‍δGTvixεGTvi=∑i=1k‍δGvi+tdGvi+txεGvi+1=xξc(G,x)+txξc(G,x)+txθ(G,x),∑i=1k‍∑j=1t‍δGTvijxεGTvij=∑i=1k‍∑j=1t‍dGvi+txεGvi+2=x2tξc(G,x)+t2x2θ(G,x).
After addition, we get(9)ξcGT,x=xξcG,x+txx+1ξcG,x+tx(tx+1)θ(G,x).It can be easily verified that expression (7) is obtained by differentiating (9) with respect to x and by putting x=1.LetGKp be the graph obtained from G by attaching t complete graphs of order p, that is, Kp, at every vertex of G. Let the vertices attached to the vertex vi be denoted by vi1(r),vi2(r),…,vip(r), i=1,2,…,k; r=1,2,…,t. Let the vertex vi be identified with vip(r), i=1,2,…,k; r=1,2,…,t.Theorem 3.
For any simple connected graphG, ξc(GKp) and ξc(G) are related as ξc(GKp)=ξc(G)+2t(p-1)ξc(G)+t(2t+1)(p-1)2θ(G)+M1(G)+6et(p-1)+kt2(p-1)2(2t+2p-3), where GKp is the graph obtained from G by attaching t complete graphs Kp at each vertex of G.Proof.
The eccentricities of the vertices ofGKp are given by εGKp(vi)=εG(vi)+1, for i=1,2,…,k, and εGKp(vij(r))=εG(vi)+2, for i=1,2,…,k; j=1,2,…,p; r=1,2,…,t.
The degree of the vertices ofGKp is given by dGKp(vi)=dG(vi)+t, for i=1,2,…,k, and dGKp(vij(r))=(p-1), for i=1,2,…,k; j=1,2,…,(p-1); r=1,2,…,t.
Thus,δGKp(vi)=δG(vi) + t(p-1)+t(p-1)2, for i=1,2,…,k, and δGKp(vij(r))=dG(vi)+t(p-1)dG(vi)+(p-1)(p-2), for i=1,2,…,k; j=1,2,…,(p-1); r=1,2,…,t.
Therefore the modified eccentric connectivity index ofGKp is given by ξc(GKp)=∑i=1kδGKp(vi)εGKp(vi)+∑i=1k∑j=1p-1∑r=1tδGKp(vij(r))εGKp(vij(r)).
Now,(10)∑i=1k‍δGKpviεGKpvi=∑i=1k‍δGvi+tp-1dGvi+tp-12kkl×εGvi+1=∑i=1k‍δGviεGvi+tp-1∑i=1k‍dGviεGvi+tp-12∑i=1k‍εG(vi)+∑i=1k‍δG(vi)+t(p-1)∑i=1k‍dG(vi)+∑i=1k‍tp-12=ξc(G)+t(p-1)ξc(G)+tp-12θ(G)+M1(G)+2et(p-1)+ktp-12,∑i=1k‍∑j=1p-1‍∑r=1t‍δGKp(vijr)εGKp(vijr)=∑i=1k‍∑j=1p-1‍∑r=1t‍dGvi+tp-1+p-1p-2×εGvi+2=ξcG+tp-1θG+p-1p-2θGkkkk+4e+2ktp-1+2kp-1p-2t(p-1)=t(p-1)ξc(G)+t2p-12θ(G)+tp-12×p-2θG+4etp-1+2kt2p-12+2ktp-12(p-2).
Adding, we get(11)ξcGKp=ξc(G)+2t(p-1)ξc(G)+t(2t+1)×p-12θ(G)+M1(G)+6et(p-1)+kt2p-12(2t+2p-3).Since forp=2, the generalized thorn graph GKp reduces to the usual thorn graph GT, Theorem 1 follows from Theorem 3 by substituting p=2.In the following, we find the modified eccentric connectivity polynomial of the graphGKp in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph G.Theorem 4.
For any simple connected graphG, the ξc(GKp,x) is given by ξc(GKp,x)=ξc(G,x)+xt(p-1)(x+1)ξc(G,x)+tx(p-1)2(1+tx+x(p-2))θ(G,x), where GKp is the graph obtained from G by attaching t complete graphs Kp at each vertex of G.Proof.
From definition, the modified eccentric connectivity polynomial ofGKp is given by
(12)ξcGKp,x=∑i=1kδGKpvixεGKpvi+∑i=1k‍∑j=1p-1‍∑r=1t‍δGKp(vij(r))xεGKp(vij(r)).
Now,(13)∑i=1k‍δGKpvixεGKpvi=∑i=1k‍δGvi+tp-1dGvi+tp-12xεGvi+1=x∑i=1k‍δGvixεGvi+tp-1∑i=1k‍dGvixεGvikkkkkk+tp-12∑i=1k‍xεGvi=xξcG,x+txp-1ξcG,x+txp-12θ(G,x).
Similarly,(14)∑i=1k‍∑j=1p-1‍∑r=1t‍δGKpvijrxεGKpvijr=∑i=1k‍∑j=1p-1‍∑r=1t‍dGvi+tp-1+p-1p-2×xεGvi+2=x2ξcG,x+tp-1θG,xkkk+p-1p-2θG,xt(p-1).
Adding the above two, we get(15)ξcGKp,x=ξc(G,x)+xt(p-1)(x+1)ξc(G,x)+txp-12(1+tx+x(p-2))θ(G,x).It can also be verified that expression (11) is obtained by differentiating (15) with respect to x and by putting x=1. Also Theorem 2 follows from Theorem 4 by substituting p=2.Let us now construct a graphGPm by attaching t paths of order m(≥2) at each vertex vi,1≤i≤k of G. The vertices of the rth path attached to vi are denoted by vi1(r),vi2(r),…,vim(r),i=1,2,…,k; r=1,2,…,t. Let the vertex vi1(r) be identified with the ith vertex vi of G. Clearly the resulting graph GPm consists of {(m-1)t+k} number of vertices.Theorem 5.
For any simple connected graphG with k vertices, ξc(GPm) and ξc(G) are related as ξc(GPm)=ξc(G)+2tξc(G)+(t2-7t+4mt)θ(G)+(m-1)M1(G)+2et(2m-1)+kt(6m2-16m+mt+9), where GPm is the graph obtained from G by attaching t paths each of length m at each vertex of G.Proof.
The eccentricities of the vertices ofGPm are given by εGPm(vi)=εG(vi)+(m-1), for i=1,2,…,k, and εGPm(vij(r))=εG(vi)+m+j-2, for i=1,2,…,k; j=1,2,…,m; r=1,2,…,t.
The degrees of the vertices ofGPm are given by dGPm(vi)=dG(vi)+t, for i=1,2,…,k; dGPm(vij(r))=2, for i=1,2,…,k; j=2,…,(m-1); r=1,2,…,t and dGPm(vim(r))=1, for i=1,2,…,k; r=1,2,…,t.
Thus,δGPm(vi)=δG(vi)+tdG(vi)+2t, for i=1,2,…,k; δGPm(vij(r))=4, for i=1,2,…,k; j=3,4,…,(m-2); r=1,2,…,t; δGPm(vi2(r))=2+dG(vi)+t, for i=1,2,…,k; r=1,2,…,t; δGP(vi(m-1)(r))=3, for i=1,2,…,k; r=1,2,…,t and δGP(vim(r))=2, for i=1,2,…,k; r=1,2,…,t.
Therefore, the modified eccentric connectivity index ofGPm is given by
(16)ξcGPm=∑i=1k‍δGPmviεGPmvi+∑i=1k‍∑j=2m‍∑r=1t‍δGPm(vij(r))εGPm(vij(r)).
Now,(17)∑i=1k‍δGPmviεGPmvi=∑i=1k‍δGvi+tdGvi+2tεGvi+m-1=∑i=1k‍δGviεGvi+t∑i=1k‍dGviεGvi+2t∑i=1k‍εGvi+m-1∑i=1k‍δGvi+tm-1×∑i=1k‍dGvi+2tkm-1=ξcG+tξcG+2tθG+m-1M1G+2et(m-1)+2kt(m-1).
Also,(18)∑i=1k‍∑j=2m‍∑r=1t‍δGPm(vijr)εGPm(vijr)=∑i=1k‍∑j=3m-2‍∑r=1t‍δGPmvijrεGPvijr+∑i=1k‍∑r=1t‍δGPmvi2rεGPmvi2r+∑i=1k‍∑r=1t‍δGPm(vim-1r)εGPm(vim-1r)+∑i=1k‍∑r=1t‍δGPmvimrεGPmvimr=∑i=1k‍∑j=3m-2‍∑r=1t‍4εGvi+m+j-2+∑i=1k‍∑r=1t‍2+dGvi+tεGvi+m+∑i=1k‍∑r=1t‍3εGvi+2m-3+∑i=1k‍∑r=1t‍2εG(vi)+2(m-1)=tξcG+t2+2tθG+2emt+mtt+2k+3tθG+3kt2m-3+2tθG+4ktm-1+4tm-4θG+4ktm-2m-4+2kt(m-1)(m-2)-6.
Combining the above, we have(19)ξcGPm=ξcG+2tξcG+t2-7t+4mtθG+m-1M1G+2et2m-1+kt6m2-16m+mt+9.Since the generalized thorn graphGPm also reduces to the usual thorn graph GT for m=2, Theorem 1 follows from Theorem 5 by substituting m=2.In the following, we find the modified eccentric connectivity polynomial of the graphGPm in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph G.Theorem 6.
For any simple connected graphG, the modified eccentric connectivity polynomial ξc(GPm,x) is given by ξc(GPm,x)=txm-1ξc(G,x)+txm-1(tx+1)ξc(G,x)+txm-1{2+x(t+2) + xm-2(3+2x)+4x2∑i=0m-5xi}θ(G,x), where GPm is the graph obtained from G by attaching t paths each of length m at each vertex of G.Proof.
The modified eccentric connectivity polynomial ofGPm is given by
(20)ξcGPm,x=∑i=1k‍δGPmvixεGPmvi+∑i=1k‍∑j=2m‍∑r=1t‍δGPm(vij(r))xεGPm(vij(r)).
Now, proceeding as above theorem, we have(21)∑i=1k‍δGPmviεGPmvi=∑i=1k‍δGvi+tdGvi+2txεGvi+m-1=xm-1∑i=1k‍δGvixεGvi+t∑i=1k‍dGvixεGvikkkkkkkkk+2t∑i=1k‍xεGvi=xm-1ξc(G,x)+tξc(G,x)+2tθ(G,x).
Again,(22)∑i=1k‍∑j=2m‍∑r=1t‍δGPmvijrxεGPmvijr=∑i=1k‍∑j=3m-2‍∑r=1t‍δGPmvijrxεGP(vijr)+∑i=1k‍∑r=1t‍δGPmvi2rxεGPm(vi2r)+∑i=1k‍∑r=1t‍δGPmvim-1rxεGPmvim-1r+∑i=1k‍∑r=1t‍δGPmvimrxεGPm(vimr)=∑i=1k‍∑j=3m-2‍∑r=1t‍4xεGvi+m+j-2+∑i=1k‍∑r=1t‍2+dGvi+txεGvi+m+∑i=1k‍∑r=1t‍3xεG(vi)+2m-3+∑i=1k‍∑r=1t‍2xεG(vi)+2(m-1)=4txm+11+x+x2+⋯+xm-5θG,x+txmt+2θG,x+xmtξcG,x+3x2m-3tθG,x+2tx2(m-1)θ(G,x).
Combining the above two, we get(23)ξcGPm,x=txm-1ξcG,x+txm-1tx+1ξcG,x+txm-1×2+x(t+2)+xm-2(3+2x)+4x2∑i=0m-5‍xiθ(G,x).Here also, differentiating (23) with respect to x and putting x=1, we get relation (19).



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