Total Eccentricity of some Graph Operations

Total Eccentricity of some Graph Operations

Khadijeh Fathalikhani 1

Department of Mathematics, Faculty of Mathematical Sciences,University of Kashan,

Kashan, Iran

Hamed Faramarzi 2, Hassan Yousefi-Azari 3

School of Mathematics, Statistics and Computer Science,University of Tehran,

Tehran, Iran

Abstract

Let G be a simple connected graph. The total eccentricity of a graph G, ζ(G),is defined as ζ(G) =

∑v∈V (G) ecG(v), where ecG(v) of a vertex v ∈ V (G) is the

maximum distance between v and any other vertex in G. In this paper, the totaleccentricity of some graph operations are computed and then a bound for that oftensor product is presented.

Keywords: Total eccentricity, graph operations.

1 Email: [email protected] Email: [email protected] Email: [email protected]

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 45 (2014) 125–131

1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

http://dx.doi.org/10.1016/j.endm.2013.11.025

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1 Introduction

In the late 1940s, one the most well-known distance-based parameter, calledWiener index, was introduced to analyze the chemical properties of paraffins[5]. Mathematical properties and chemical applications of this index have beenextensively researched. Many other indices have been defined. One of therecent ones is the eccentric connectivity index. This index may provide leadsfor the development of safe anti-HIV compounds [3]. For more informationabout distance-based invariants and the eccentric connectivity index, see [1],[4] . An other parameter is derived from the eccentric connectivity index:total eccentricity. The aim of this paper is to investigate some mathematicalproperties of total eccentricity of some graph operations.

Consider a simple connected graph G, and let V (G) and E(G) denote itsvertex and edge sets, respectively. |V (G)| = n(G) is called the order of G. Thedistance between u and v in V (G), dG(u, v), is the length of the shortest u−vpath in G. The eccentricity of a vertex v ∈ V (G), ecG(v), is the maximumdistance between v and any other vertex in G. For the vertex v in K1, wedefine ecG(v) = 0. The degree of a vertex v ∈ V (G), degG(v), is the number ofedges incident to vertex v. The eccentric connectivity index of G is defined asξc(G) =

∑v∈V (G) ecG(v)degG(v). Moreover, The total eccentricity of a graph

G, ζ(G), is defined as ζ(G) =∑

v∈V (G) ecG(v). The diameter of G, D, is the

maximum value of the eccentricities of the vertices of G. A vertex v ∈ V (G)is called well-connected if degG(v) = |V (G)| − 1, i.e., if it is adjacent to anyother vertex in G.

The Cartesian product of two graphs G and H , G × H , has vertex setV (G×H) = V (G)× V (H) and (a, x)(b, y) is an edge of G×H if a = b andxy ∈ E(H), or ab ∈ E(G) and x = y. The join G+H of graphs G and H is agraph with the vertex set V (G)∪V (H) and the edge set E(G)∪E(H)∪{uv :u ∈ V (G) and v ∈ V (H)}. The composition G[H ] of two graphs G and Hwith disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) is agraph with vertex set V (G) × V (H) and u = (a, x) is adjacent to v = (b, y)whenever a is adjacent to b or a = b and x is adjacent to y. The symmetricdifference G ⊕ H of two graphs G and H is the graph with the vertex setV (G) × V (H) and E(G ⊕ H) = {a, x)(b, y)|ab ∈ E(G) or xy ∈ E(H) butnot both}. The tensor product G ⊗ H of two graphs G and H is the graphwith vertex set V (G) × V (H) and E(G ⊗ H) = {(a, x)(b, y)|ab ∈ E(G) andxy ∈ E(H)}.

In the whole paper, graphs have been assumed to be simple and connected,unless stated otherwise. Moreover, if there is no ambiguity, then the subscript

K. Fathalikhani et al. / Electronic Notes in Discrete Mathematics 45 (2014) 125–131126

G may be omitted.

2 Main Results

Let us begin with a few examples.

Example 2.1 For a cycle graph with n vertices, we have:

ζ(Cn) =

{n2/2 if n is even

n(n− 1)/2 if n is odd

Example 2.2 Let Qn be the cypercube graph on n vertices. Then, ζ(Qn) =n2n.

Now we provide a bound:

Theorem 2.3 Let G = (V,E) be a connected graph of order n and diameterD, then n ≤ ζ(G) ≤ nD, and the bounds are sharp.

Proof. Let A be the set of all well-connected vertices and let B be the set ofall vertices of G whose degrees is less than or equal to n − 2. Then ζ(G) =∑

v∈A ec(v) +∑

v∈B ec(v) ≥ |A| + |B| = n. The proof for the upper bound isthe same. It is clear that Kn, the complete graph on n vertices, and Cn, thecycle on n vertices, give the bounds. �

Here we bring some properties of the graph operations:

Theorem 2.4 Let G and H be two graphs. Then we have:

(a) |V (G×H)| = |V (G⊗H)| = |V (G[H ])| = |V (G⊕H)| = |V (G)|.|V (H)|and |E(G×H)| = |E(G)|.|V (H)|+ |V (G)|.|E(H)|

(b) |E(G1⊕...⊕Gn)| =n∑

n′=1

(−4)n′−1

n−1∑j1=1

n∑j2=1

...n∑

jn′=1

(n∏

i=1,i �=j1,...jn′V 2i )(

n′∏k=1

Ejk)+

En

n−1∏i=1

V 2i , where Ei = |E(Gi)| and Vi = |V (Gi)|.

(c) If (a, x) and (b, y) are vertices of G × H then dG×H((a, x), (b, y)) =dG(a, b) + dH(x, y)

Proof. (a) is obvious by definition. The proof of (b) is by an induction andfor (c), we refer the reader to [6]. �

The following remark is derived from [2].

K. Fathalikhani et al. / Electronic Notes in Discrete Mathematics 45 (2014) 125–131 127

Remark 2.5 Let G1 and G2 be two graphs. If G = G1 × G2 then for everyvertex (a, x) ∈ V (G),

degG(a, x) = degG1(a) + degG2(x),

and

ecG(a, x) = ecG1(a) + ecG2(x).

Lemma 2.6 Let G1 and G2 be two graphs. If G = G1 ×G2, then:

ζ(G) = |V (G2)|ζ(G1) + |V (G1)|ζ(G2).

Proof. Using the definition, we have:

ζ(G)=∑

(a,x)∈V (G)

ecG((a, x))

=∑

(a,x)∈V (G)

(ecG1(a) + ecG2(x))

=∑

a∈V (G1)

∑x∈V (G2)

(ecG1(a) + ecG2(x))

= |V (G2)|ζ(G1) + |V (G1)|ζ(G2).

�

By a simple induction, we have the following lemma:

Lemma 2.7 Let G1, ..., Gn be n graphs. If G = G1 × ...×Gn, then:

ζ(G) =n∑

i=1

ζ(Gi)n∏

j=1,j �=i

|V (Gj)|.

Proof. By applying an induction argument and the previous lemma, the re-sult is obtained. �

In particular, we have ζ(Gn) = nζ(G)|V (G)|n−1, whichGn denotesG× ...×G︸︷︷︸n times

.

Theorem 2.8 Let G1 and G2 be two graphs. If G = G1 × G2, then ξc(G) =n2ξ

c(G1) + n1ξc(G2) + 2m1ζ(G2) + 2m2ζ(G1), where ni = |V (Gi)| and mi =

|E(Gi)|.Proof. For the proof we refer the reader to [2]. �

Corollary 2.9 Let G1, ..., Gn be n graphs and G = G1 × ...×Gn. Then,

ξc(G) =n∑

i=1

ξc(Gi)n∏

j=1,j �=i

|V (Gj)|+2n∑

k=1

[n∑

i=1,i �=k

|E(Gi)|n∏

j=1,j �=i,k

|V (Gj)|ζ(Gk),

and in particular,


ξc(Gn) =n∑

i=1

ξc(G)|V (G)|n−1 + 2n(n− 1)|E(G)||V (G)|n−2ζ(G).

In what follows we present some elementary lemmas which are crucial forcomputing the total eccentricity of composition of two graphs.

Lemma 2.10 Let G1 and G2 be two graphs. If G = G1[G2] then for everyvertex (a, x) ∈ V (G), degG(a, x) = degG1(a)|V (G2)|+ degG2(x).

Proof. By the definition, the proof is straight forward. �

Lemma 2.11 Let G1 and G2 be two graphs. If G1 does not have any well-connected vertex, then ecG1[G2](a, x) = ecG1(a).

Proof. The proof can be found in [2]. �

Theorem 2.12 Let G1 and G2 be two graphs. If G1 does not contain anywell-connected vertex and G = G1[G2], then

ζ(G) = |V (G2)|ζ(G1).

Proof. Using the definition, we write:

ζ(G)=∑

(a,x)∈V (G)

ecG(a, x)

=∑

a∈V (G1)

∑x∈V (G2)

ecG1(a)

= |V (G2)|ζ(G1).

�

Theorem 2.13 Let G1 and G2 be two graphs. If G1 does not contain anywell-connected vertex and G = G1[G2], then

ξc(G) = ξc(G1)|V (G2)|2 + 2|E(G2)|ζ(G1).

Proof. For the proof we refer to [2]. �

Lemma 2.14 Let G1 and G2 be two graphs, then:

degG1⊕G2(a, x) = degG1(a)[|V (G2)|−degG2(x)]+degG2(x)[|V (G1)|−degG1(a)].

Proof. The proof is straightforward and so it is omitted. �

Two following two results are derived from [2].

Lemma 2.15 Let G1 and G2 be two graphs. If none of them contains anywell-connected vertex, then ecG1⊕G2(a, x) = 2.


Theorem 2.16 Let G1 and G2 be two graphs. If none of them contains anywell-connected vertex, then

ξc(G1 ⊕G2) = 4(V 22 E1 + V 2

1 E2 − 4E1E2),

where Vi = |V (Gi)| and Ei = |E(Gi)|.Corollary 2.17 Let G1 and G2 be two graphs. If none of them contains anywell-connected vertex, then

ζ(G1 ⊕G2) = 2|V (G1)||V (G2)|.Proof. Using two previous results and definition, the proof is straightfor-ward. �

Lemma 2.18 Let G1 and G2 be two graphs. Then,

(a) degG1⊗G2(a, x) = degG1(a)degG2(x).

(b) If at least one of them is not bipartite and G = G1 ⊗G2, then

dG((a, x), (b, y)) ≥ max{dG1(a, b), dG2(x, y)}.Proof. The first part can be obtained by the definition. For the second part,we should notice that the tensor product of G1 and G2 is a subgraph of theirstrong product. �

Lemma 2.19 Let G1 and G2 be two graphs that at least one of them is notbipartite. If G = G1 ⊗G2, then:

ecG(a, x) ≥ max{ecG1(a), ecG2(x)}.Theorem 2.20 Let G1 and G2 be two graphs that at least one of them is notbipartite. Then,

ζ(G1 ⊗G2) ≥∑

a∈V (G1)

∑x∈V (G2)

max{ecG1(a), ecG2(x)},and

ξc(G1 ⊗G2) ≥∑

a∈V (G1)

∑x∈V (G2)

degG1(a)degG2(x)max{ecG1(a), ecG2(x)}.Proof. The proof follows from Lemmas 2.18 and 2.19. �

References

[1] A. R. Ashrafi, M. Saheli, M. Ghorbani, The eccentric connectivity index ofnanotubes and nanotori, Journal of Computational and Applied Mathematics235 (2011), 4561–4566.


[2] T.Doslic, M. Ghorbani, M. A. Hosseinzadeh Eccentric connectivity polynomialof some graph operations, Util. Math. 84 (2011), 297-309.

[3] H. Dureja, S. Gupta, A.K. Madan, Predicting anti-HIV-1 activity of 6-arylbenzonitriles: Computational approach using superaugmented eccentricconnectivity topochemical indices, Journal of Molecular Graphics and Modelling26 (2008), 1020–1029.

[4] M.J. Morgan, S. Mukwembi, H.C. Swart, On the eccentric connectivity index ofa graph, Discrete Mathematics 311-13 (2010), 1229–1234.

[5] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem.Soc. 69 (1947), 17–20.

[6] Y-N. Yeh, I. Gutman On the sum of all distances in composite graphs, DiscreteMath. 135 (1994), 17–20.


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