검색
Article Search

JMB Journal of Microbiolog and Biotechnology

OPEN ACCESS eISSN 0454-8124
pISSN 1225-6951
QR Code

Article

Kyungpook Mathematical Journal 2021; 61(1): 61-74

Published online March 31, 2021

Copyright © Kyungpook Mathematical Journal.

Distance Eccentric Connectivity Index of Graphs

Akram Alqesmah, Anwar Saleh, R. Rangarajan, Aysun Yurttas Gunes and Ismail Naci Cangul∗

Department of Studies in Mathematics, University of Mysore, Mysore 570006, India
e-mail: aalqesmah@gmail.com

Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
e-mail: asaleh1@uj.edu.sa

Department of Studies in Mathematics, University of Mysore, Mysore 570006, India
e-mail: rajra63@gmail.com

Bursa Uludag University, Mathematics, Gorukle 16059 Bursa-Turkey
e-mail: ayurttas@uludag.edu.tr and cangul@uludag.edu.tr

Received: July 6, 2019; Revised: April 14, 2020; Accepted: May 18, 2020

Let G=(V,E) be a connected graph. The eccentric connectivity index of G is defined by ξC(G)= uV(G)deg(u)e(u), where deg(u) and e(u) denote the degree and eccentricity of the vertex u in G, respectively. In this paper, we introduce a new formulation of ξC that will be called the distance eccentric connectivity index of G and defined by ξDe(G)= uV(G)degDe(u)e(u) where degDe(u) denotes the distance eccentricity degree of the vertex u in G. The aim of this paper is to introduce and study this new topological index. The values of the eccentric connectivity index is calculated for some fundamental graph classes and also for some graph operations. Some inequalities giving upper and lower bounds for this index are obtained.

Keywords: eccentric connectivity index, distance eccentric connectivity index, topological graph index, graph operation.

In this paper, we are concerned with only connected simple graphs G=(V,E) which are finite, undirected with no loops nor multiple edges and for any two vertices u and v in G, there exists a uv-path starting from u and ending in v. Throughout this paper, we take p=|V(G)| and q=|E(G)|. The complement of G, denoted by G¯, is a simple graph on the same set of vertices V(G) in which two vertices u and v are adjacent in G¯ if and only if they are not adjacent in G. The distance between any two vertices u and v in G denoted by d(u,v) is the number of edges on a shortest path joining u and v. There are many interesting research areas in graph theory concentrate on the distance. Many well known graph topological indices, the most famous one being Wiener index, are defined in terms of distances between the vertices of a given graph. The eccentricity e(u) of a vertex u in G is another distance based concept and defined as the maximum distance between u and any other vertex v in G, that is e(u)=max{d(u,v):vV(G)}. This notion plays an important role in the study of graphs and has many applications. The maximum and minimum eccentricity over all vertices of G are called the diameter diam(G) and the radius rad(G) of G, respectively. All the definitions and terminologies about graphs used in this paper are available in [13]. The path, wheel, cycle, star and complete graphs with p vertices are denoted by Pp, Wp, Cp, Sp and Kp, respectively, and Kr,s is the complete bipartite graph on r+s vertices.

In chemical graph theory, various graph invariants are used for establishing correlations of chemical structures with various physical properties, chemical reactivity, or biological activity. These graph invariants are called topological indices of (molecular) graphs. Although most of the topological graph indices are degree, distance or matrix based, there are some eccentricity-based topological indices in chemical graph theory.

In [11], the authors introduced a topological descriptor called the connectivity eccentricity index when investigating the antihypertensive activity of derivatives of N-benzylimidazole. They showed that the results obtained using the connectivity eccentricity index were better than the corresponding values obtained using Balaban's mean square distance index, [5, 6], and the accuracy of prediction was found to be about 80 percent of the active range, [11]. In [22], the authors introduced the eccentric connectivity index of a given graph, which has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature, [12, 21, 22]. The eccentric connectivity index ξC(G) of a graph G is defined as ξC(G)= uV(G)deg(u)e(u), [22]. When the vertex degrees are not taken into account, we obtain the total eccentricity of the graph G as ζ(G)= uV(G)e(u). We refer the reader to [3, 4, 10, 15, 17, 19, 18, 24] for explicit explanations of the related notions to eccentric connectivity index of graphs. The relation between the eccentricity indices and other topological indices has also been investigated. In [14], the relationship between eccentric connectivity index and Zagreb indices were studied and in [7], the Zagreb and multiplicative Zagreb indices of some graphs were calculated. In [20], the relation of the notion of eccentricity and other notions related to distance in graphs are correlated with the edge fixed geodomination number of a graph.

Let uV(G). The distance eccentricity neighborhood of u denoted by NDe(u) is defined as

NDe(u)={vV(G):d(u,v)=e(u)}.

The cardinality of NDe(u) is called the distance eccentricity degree of the vertex u in G and denoted by degDe(u), and NDe[u]=NDe(u){u} will be called as the closed distance eccentricity neighborhood of u. Note that if u has a full degree in G, then deg(u)=degDe(u) and if e(u)=2, then degG¯(u)=degDe(u). We use the symbol Q(G) to denote the sum of distance eccentricity degrees of all vertices in a connected graph G, that is

Q(G)= uV(G)degDe(u).

The maximum and minimum distance eccentricity degrees of a vertex in G are denoted respectively by ΔDe(G) and δDe(G):

ΔDe(G)=maxuV|NDe(u)|andδDe(G)=minuV|NDe(u)|,

see [2]. Also, we denote the set of vertices of G with eccentricity equal to α by Veα(G)V(G), where α=1,2,,diam(G).

The applications of vertex eccentricity and the eccentric connectivity index of graphs motivated us to use the distance eccentricity degree instead of the normal degree which is very closed to the vertex eccentricity.

In this paper, we study this topological index by investigating some of its properties and its relations withξC(G) and ζ(G) of a connected graph G. One of the ways of studying graphs is graph operations. There is a number of such operations which help us to determine the properties of a large graph in terms of similar properties of the relatively smaller component graphs. For some examples, see [1, 8, 9, 16]. We compute the exact values of the distance eccentric connectivity index for some specific graphs and graph operations. Some upper and lower bounds and interesting results are obtained.

In this section, we define the distance eccentric connectivity index ξDe(G) of a connected graph G and compute its exact values for some specific graphs. Also, we give some bounds on ξDe and its relation with some other topological indices of graph.

Definition 2.1.

Let G=(V,E) be a connected graph. The distance eccentric connectivity index of G is defined by

ξDe(G)= uV(G)degDe(u)e(u),

where degDe(u) denotes to the distance eccentricity degree of the vertex u in G.

The distance eccentric connectivity index ξDe(G) of some well-known graph classes are as follows:

Proposition 2.2.

  • (i) For p2, ξDe(Kp)=p(p1).

  • (ii) For p2, ξDe(Pp)=p2(2pp21).

  • (iii) For p3, ξDe(Cp)=p22, piseven;2pp2, pisodd.

  • (iv) For r,s2, ξDe(Kr,s)=2(r(r1)+s(s1)).

  • (v) For p3, ξDe(Sp)=(p1)(2p3).

  • (vi) For p5, ξDe(Wp)=(p1)(2p7).

In the following, we try to determine some general bounds on ξDe for a graph G and its complement G¯.

Proposition 2.3.

Let G be a connected graph on p vertices and q edges. Then

ξDe(G)|Ve1(G)|(p1)+2q¯diam(G),

where Ve1(G)V(G) is the set of vertices of eccentricity one in G and q¯ is the number of edges in the complement G¯ of G. The equality holds if and only if diam(G)2.

Proof. Let G be a connected graph and u ∈ V(G). If uVe1(G), then degDe(u)=deg(u)=p1 and if uV=V(G)Ve1(G), then degDe(u)degG¯(u). Thus

ξDe(G)= uV(G)degDe(u)e(u)=|Ve1(G)|(p1)+ uVdegDe(u)e(u)|Ve1(G)|(p1)+diam(G) uVdegG¯(u)=|Ve1(G)|(p1)+2q¯diam(G).

For the equality, it is clear that degDe(u)=degG¯(u), for every u ∈ V if and only if diam(G)2. Hence, the result holds.

Corollary 2.4.

Let G be a (p,q)-connected graph such that Ve1(G)=ϕ (G has no vertex of full degree). Then

prad(G)ξDe(G)2q¯diam(G).

Theorem 2.5.

Let G be a (p,q)-connected graph with diam(G)4 and G¯ be also connected. Then

ξDe(G¯)=4q.

Proof. It is well known that if a graph G is connected with diam(G)4 and its complement G¯ is also connected, then diam(G¯)2. Also, since G and G¯ are both connected, then eG¯(u)=2, for every uV(G). Therefore, degG¯De(u)=degG(u), for every uV(G). Hence,

  ξDe(G¯)= uV(G)deg G ¯ De(u)e G ¯ (u)=2 uV(G)degG(u)=4q.

Proposition 2.6.

For any (p,q)-connected graph G, we have

ξDe(G)(p1)ζ(G)ξc(G)+|Ve1(G)|(p1).

Furthermore, the equality is attained if and only if diam(G)2.

Proof. Suppose first G has no vertices of full degree (Ve1(G)=ϕ). Then

degGDe(u)deg G¯(u),

for all uV(G). Therefore,

ξDe(G) uV(G)(p1degG(u))e(u)=(p1)ζ(G)ξc(G).

Suppose now Ve1(G)ϕ. Then

ξDe(G)|Ve1(G)|(p1) uV(G)(p1degG(u))e(u)=(p1)ζ(G)ξc(G),

which is the required result.

For the equality, it is clear that degDe(u)=degG¯(u), for all u ∈ V if and only if diam(G)2.

The first Zagreb eccentricity and the first distance eccentricity Zagreb indices of a connected graph G were defined respectively as

ζ1(G)= uV(G)(e(u))2

and

M1De(G)= uV(G)(degDe(u))2,

[2, 23]. In the following two propositions we give relations between the distance eccentric connectivity index ξDe(G), the first Zagreb eccentricity index ζ1(G) and the first distance eccentricity Zagreb index M1De(G) of a connected graph G.

Proposition 2.7.

Let G be a connected graph on p2 vertices. Then

ξDe(G)pζ(G)ζ1(G),

with the equality if and only if GKp or GSp.

Proof. For u ∈ V(G), let d(u;i) be the number of vertices that have distance i from u in G, where i=1,2,,e(u). Then we have

p1=degDe(u)+ i=1 e(u)1d(u;i)degDe(u)+ i=1 e(u)11=degDe(u)+e(u)1.

Therefore, degDe(u)pe(u) with the equality if and only if e(u)=1 or e(u)2 with d(u;1)=d(u;2)==d(u;e(u)1)=1. Then

ξDe(G)= uV(G)degDe(u)e(u) uV(G)(pe(u))e(u)=pζ(G)ζ1(G).

Suppose the equality holds. Then degDe(u)=pe(u) and hence either e(u)=1 or e(u)2 with d(u;1)=d(u;2)==d(u;e(u)1)=1.

Case 1. Suppose e(u)=1 for some u ∈ V(G). Then degDe(u)=p1 and thus e(v)=1 or 2 for every vu in G. Now, if e(v)=1 for all vu, then GKp and if e(v)=2 for some vu with d(v;1)=1 and thus the vertex u is unique for all vu and hence GSp.

Case 2. Suppose now e(u)2 with d(u;1)=d(u;2)==d(u;e(u)1)=1 for all u ∈ V(G). Let diam(G)=r and P(G)=u1u2ur be the diametral path in G. Clearly d(u2;1)>1, a contradiction. Hence there is no connected graph G satisfying the condition e(u)2 with d(u;1)=d(u;2)==d(u;e(u)1)=1 for all uV(G). The converse is clear.

Proposition 2.8.

Let G, G¯ be connected graphs on p vertices and q, q¯ edges, respectively. Then

ξDe(G)2pq¯M1De(G).

Proof. From the proof of Proposition 2.7, we have for any u ∈ V(G), e(u)pdegDe(u) with the equality if and only if e(u)=1 or e(u)2 with d(u;1)=d(u;2)==d(u;e(u)1)=1. Since G and G¯ are both connected, then degDe(u)degG¯(u) for all u ∈ V(G). Therefore,

ξDe(G) uV(G)degDe(u)(pdegDe(u))  =p uV(G)degDe(u)M1De(G)  2pq¯M1De(G)

In this section, we compute the distance eccentric connectivity index of some graph operations.

Cartesian Product

The cartesian product of two graphs G1 and G2 where |V(G1)|=p1, |V(G2)|=p2 and |E(G1)|=q1, |E(G2)|=q2 is denoted by G1G2. It has the vertex set V(G1)×V(G2) and two vertices (u,u) and (v,v) are connected by an edge if and only if either ([u=v and uvE(G2)]) or ([u'=v' and uvE(G1)]). In other words, |E(G1G2)|=q1p2+q2p1. Therefore the degree of a vertex (u,u) of G1G2 is as follows:

degG1G2(u,u)=degG1(u)+degG2(u).

The cartesian product of more than two graphs is similarly denoted by

i=1nGi=G1G2Gn=(G1G2Gn1)Gn,

in which any two vertices u=(u1,u2,,un) and v=(v1,v2,,vn) are adjacent in i=1nGi if and only if ui=vi, ij and ujvjE(Gj), where i,j=1,2,,n. If G1=G2==Gn=G, we have the n-th cartesian power of G and denote it by Gn.

Lemma 3.1.

Let G= i=1nGi and let u=(u1,u2,,un) be a vertex in V(G). Then

e(u)= i=1ne(ui).

Lemma 3.2.

Let G= i=1nGi and let u=(u1,u2,,un) be a vertex in G. Then

degGDe(u)= i=1ndegGiDe(ui).

As a result we have

Theorem 3.3.

Let G= i=1nGi. Then

ξDe(G)= i=1nj=1jinξDe(Gi)Q(Gj).

Proof. Let u=(u1,u2,,un)be a vertex in V(G). Then by Lemmas 3.1 and 3.2, we have

ξDe(G)= uV(G)degGDe(u)eG(u)= (u1 ,,un )V(G) i=1ndegGiDe(ui) i=1ne(ui)= u1 V( G1 ) u2 V( G2 ) un V( Gn ) i=1ndegGiDe(ui)(e(u1)++e(un))= i=1n j=1 ji nξDe(Gi)Q(Gj).

Composition

The composition G=G1[G2] of two graphs G1 and G2 with disjoint vertex sets V(G1) and V(G2) and edge sets E(G1) and E(G2), where |V(G1)|=p1, |E(G1)|=q1 and |V(G2)|=p2, |E(G2)|=q2 is the graph with vertex set V(G1)×V(G2) where any two vertices (u,u') and (v,v') are adjacent whenever u is adjacent to v in G1 or u=v and u' is adjacent to v' in G2. Thus

|E(G1[G2])|=q1p22+q2p1.

The degree of a vertex (u,u') of G1[G2] is as follows:

degG1[G2](u,u)=p2degG1(u)+degG2(u).

Lemma 3.4.

Let G=G1[G2] and e(v)1 for all vV(G1). Then eG((u,u))=eG1(u).

Lemma 3.5.

Let G=G1[G2] and e(v)1 for all vV(G1). Then

degGDe(u,u)=p2degG1De(u)+degG2¯(u), ifuVe2(G1);p2degG1De(u),otherwise.

Theorem 3.6.

Let G=G1[G2] and e(v)1 for all vV(G1). Then

ξDe(G)=p22ξDe(G1)+4q2¯|Ve2(G1)|.

Proof. By Lemmas 3.4 and 3.5, we have

ξDe(G)= (u,u )V(G)degGDe(u,u)eG(u,u)  = uV( G1 ) uV( G2)degGDe(u,u)eG(u,u)  = uV e2 ( G1 ) uV( G2)(p2degG 1De(u)+degG 2 ¯(u))eG 1(u)  + uV( G1 )V e2 ( G1 ) uV( G2)(p2degG 1De(u))eG 1(u)  = uV( G1 ) uV( G2)(p2degG 1De(u))eG 1(u)  + uV e2 ( G1 ) uV( G2)degG 2 ¯(u)eG 1(u)  =p22ξDe(G1)+4q2¯|Ve2(G1)|.

Corollary 3.7.

Let G=G1[G2] and e(v)1 or 2 for all vV(G1). Then

ξDe(G)=p22ξDe(G1).

Disjunction and Symmetric Difference

The disjunction G1G2 of two graphs G1 and G2 with |V(G1)|=p1, |E(G1)|=q1 and |V(G2)|=p2, |E(G2)|=q2 is the graph with vertex set V(G1)×V(G2) in which (u,u') is adjacent to (v,v') whenever u is adjacent to v in G1 or u' is adjacent to v' in G2. So,

|E(G1G2)|=q1p22+q2p122q1q2.

The degree of a vertex (u,u') of G1G2 is as follows:

degG1G2(u,u)=p2degG1(u)+p1degG2(u)degG1(u)degG2(u).

Also, the symmetric difference G1G2 of G1 and G2 is the graph with vertex set V(G1)×V(G2) in which (u,u') is adjacent to (v,v') whenever u is adjacent to v in G1 or u' is adjacent to v' in G2, but not both. From definition one can see that,

|E(G1G2)|=q1p22+q2p124q1q2.

The degree of a vertex (u,u') of G1G2 is as follows:

degG1G2(u,u)=p2degG1(u)+p1degG2(u)2degG1(u)degG2(u).

The distance between any two vertices of a disjunction or a symmetric difference cannot exceed two. Thus, if e(v)1 for all vV(G1)V(G2), the eccentricity of all vertices is constant and equal to two.

Lemma 3.8.

  • (i) degG1G2De(u,u)=degG1G2¯(u,u)

  • (ii) degG1G2De(u,u)=degG1G2¯(u,u).

Proposition 3.9.

Let G1 and G2 be two graphs with e(v)1 for all vV(G1)V(G2). Then

  • (i) ξDe(G1G2)=4qG1 G2 ¯,

  • (ii) ξDe(G1G2)=4qG1 G2 ¯.

Proof. The proof is straightforward.

Join

The join G1+G2 of two graphs G1 and G2 with disjoint vertex sets V(G1) and V(G2) and edge sets E(G1) and E(G2) such that |V(G1)|=p1, |V(G2)|=p2, |E(G1)|=q1 and |E(G2)|=q2 is the graph on the vertex set V(G1)V(G2) and the edge set E(G1)E(G2){u1u2:u1V(G1);u2V(G2)}. Hence, the join of two graphs is obtained by connecting each vertex of one graph to each vertex of the other graph, while keeping all edges of both graphs. The degree of any vertex uG1+G2 is given by

degG1+G2(u)=degG1 (u)+p2 , ifuV(G1 );degG2 (u)+p1 , ifuV(G2 ).

Actually, by using the definition of the join graph G= i=1nGi, one can see that the eccentricity of any vertex u∈ G does not exceed two. Therefore, if Ve1(Gi)=ϕ for all GiG, then e(u)=2 for all u ∈ G.

Lemma 3.10.

Let G= i=1nGi and uV(G). Then

degGDe(u)=|V(G)|1, uVe1(Gi);pi1degGi(u), uV(Gi)Ve1(Gi),fori=1,2,,n.

Theorem 3.11.

Let G= i=1nGi. Then

ξDe(G)=(|V(G)|1) i=1n|Ve1(Gi)|+4 i=1nqi¯.

Proof. By Lemma 3.10, we get

ξDe(G)= uV(G)degGDe(u)eG(u)  = i=1n uV( Gi)degGDe(u)eG(u)  = i=1n uV e1( Gi)(|V(G)|1)  +2 i=1n uV( Gi)V e1( Gi)(pi1degGi(u))  =(|V(G)|1) i=1n|Ve1(Gi)|+4 i=1nqi¯.

Corollary 3.12.

If Gi has no vertices of full degree (Ve1(Gi)=ϕ) for all i=1,2,,n, then

ξDe( i=1nGi)=4 i=1nqi¯.

Corona Product

The corona product G1G2 of two graphs G1 and G2 where |V(G1)|=p1, |V(G2)|=p2 and |E(G1)|=q1, |E(G2)|=q2 is the graph obtained by taking |V(G1)| copies of G2 and joining each vertex of the i-th copy with vertex u ∈ V(G1). Obviously,

|V(G1G2)|=p1(p2+1)

and

|E(G1G2)|=q1+p1(q2+p2).

It follows from the definition of the corona product G1G2 that the degree of each vertex uG1G2 is given by

degG1G2(u)=degG1 (u)+p2 , ifuV(G1 );degG2 (u)+1, ifuV(G2 ).

Lemma 3.13.

Let G=G1G2 be a connected graph such that G1K1 and let uV(G). Then

degGDe(u)=p2degG1De(u),uV(G1);p2degG1De(v),uV(G)V(G1),

where v ∈ V(G1) is adjacent to u.

Lemma 3.14.

Let G=G1G2 be a connected graph such that G1K1 and let uV(G). Then

eG(u)=eG1(u)+1, uV(G1);eG1(v)+2, uV(G)V(G1),

where v ∈ V(G1) is adjacent to u.

Theorem 3.15.

Let G=G1G2 be a connected graph such that G1K1. Then

ξDe(G)(G)=p2[(p2+1)ξDe(G1)+(2p2+1)Q(G1)].

Proof. By Lemmas 3.13 and 3.14, we have

ξDe(G)(G)= uV(G)degGDe(u)eG(u)  = uV( G1 )degGDe(u)eG(u)+ vV( G1 ) uV( G2)degGDe(u)eG(u)  = uV( G1 )p2degG1De(u)(eG1(u)+1)  + vV( G1 ) uV( G2)p2degG 1De(v)(eG 1(v)+2)  =p2[(p2+1)ξDe(G1)+(2p2+1)Q(G1)].

Example 3.16.

  • (i) For any cycle Cp1 and any path Pp2 with p13 and p21, ξDe(Cp1Pp2)=2p1 p2 [(p2 +1) p1 2 +2p2 +1],ifp1 isodd; p1 p2 2[p1 (p2 +1)+4p2 +2)],ifp1 iseven.

  • (ii) For any two cycles Cp1 and Cp2 with p1,p23, ξDe(Cp1Cp2)=2p1 p2 [(p2 +1) p1 2 +2p2 +1],ifp1 isodd; p1 p2 2[p1 (p2 +1)+4p2 +2)],ifp1 iseven.

AA and AS has proposed the problem and constructed the propositions' statements. RR has done calculations with indices. AYG and INC has obtained the inequalities and also made calculations and proofs of the results on graph operations.

  1. N. Akgunes, K. C. Das, A. S. Cevik, and I. N. Cangul. Some properties on the lexicographic product of graphs obtained by monogenic semigroups, J. Inequal. Appl. 2013, 238 (2013). 9 pp.
    CrossRef
  2. A. Alqesmah, A. Alwardi, and R. Rangarajan. On the distance eccentricity Zagreb indices of graphs, International J. Math. Combin. 4, 110-120 (2017).
  3. A. R. Ashrafi, T. Došlić, and M. Saheli. The eccentric connectivity index of T UC4C8(R) nanotubes, MATCH Commun. Math. Comput. Chem. 65(1), 221-230 (2011).
  4. A. R. Ashrafi, M. Saheli, and M. Ghorbani. The eccentric connectivity index of nanotubes and nanotori, J. Comput. Appl. Math. 235(16), 4561-4566 (2011).
    CrossRef
  5. A. T. Balaban. Highly discriminating distance-based topological index, Chem. Phys. Lett. 89(5), 399-404 (1982).
    CrossRef
  6. A. T. Balaban. Topological indices based on topological distances in molecular graph, Pure Appl. Chem. 55(2), 199-206 (1983).
    CrossRef
  7. K. C. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, and A. S. Cevik. On the first Zagreb index and multiplicative Zagreb coindices of graphs, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 24(1), 153-176 (2016).
    CrossRef
  8. K. C. Das, K. Xu, I. N. Cangul, A. S. Cevik, and A. Graovac. On the Harary index of graph operations, J. Inequal. Appl. 2013, 339 (2013). 16 pp.
    CrossRef
  9. K. C. Das, A. Yurttas, M. Togan, A. S. Cevik, and I. N. Cangul. The multiplicative Zagreb indices of graph operations, J. Inequal. Appl. 2013, 90 (2013). 14 pp.
    CrossRef
  10. T. Došlić, M. Saheli, and D. Vukiéević. Eccentric connectivity index: extremal graphs and values, Iran. J. Math. Chem. 1(2), 45-56 (2010).
  11. S. Gupta, M. Singh, and A. K. Madan. Connective eccentricity index: a novel topological descriptor for predicting biological activity, J. Mol. Graph. Model. 18(1), 18-25 (2000).
    CrossRef
  12. S. Gupta, M. Singh, and A. K. Madan. Application of graph theory: relationship of eccentric connectivity index and Wiener's index with anti-inflammatory activity, J. Math. Anal. Appl. 266(2), 259-268 (2002).
    CrossRef
  13. F. Harary, Graph theory (Addison-Wesley, Reading Mass, 1969).
    CrossRef
  14. H. Hua and K. C. Das. The relationship between eccentric connectivity index and Zagreb indices, Discrete Appl. Math. 161, 2480-2491 (2013).
    CrossRef
  15. A. Ilič and I. Gutman. Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem. 65, 731-744 (2011).
  16. E. G. Karpuz, K. C. Das, I. N. Cangul, and A. S. Cevik. A new graph based on the semi-direct product of some monoids, J. Inequal. Appl. 2013, 118 (2013). 8 pp.
    CrossRef
  17. M. J. Morgan, S. Mukwembi, and H. C. Swart. On the eccentric connectivity index of a graph, Discrete Math. 311, 1229-1234 (2011).
    CrossRef
  18. M. J. Morgan, S. Mukwembi, and H. C. Swart. A lower bound on the eccentric connectivity index of a graph, Discrete Appl. Math. 160, 248-258 (2012).
    CrossRef
  19. M. Saheli and A. R. Ashrafi. The eccentric connectivity index of armchair polyhexnanotubes, Maced. J. Chem. Chem. Eng. 29(1), 71-75 (2010).
    CrossRef
  20. A. P. Santhakumaran and P. Titus. The edge fixed geodomination number of a graph, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17(1), 187-200 (2009).
  21. S. Sardana and A. K. Madan. Application of graph theory: relationship of antimy-cobacterial activity of quinolone derivatives with eccentric connectivity index and Za-greb group parameters, MATCH Commun. Math. Comput. Chem. 45, 35-53 (2002).
  22. V. Sharma, R. Goswami, and A. K. Madan. Eccentric connectivity index: a Novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci. 37(2), 273-282 (1997).
    CrossRef
  23. R. Xing, B. Zhou, and N. Trinajstic. On Zagreb eccentricity indices, Croat. Chem. Acta 84(4), 493-497 (2011).
    CrossRef
  24. B. Zhou and Z. Du. On eccentric connectivity index, MATCH Commun. Math. Comput. Chem. 63, 181-198 (2010).