Abstract If G is a connected graph with vertex set V , then the eccentric connectivity index of G, ξ C (G), is defined as is the degree of a vertex v and ecc(v) is its eccentricity. We obtain exact formulas for calculating the eccentric connectivity index of dendrimers. Mathematics Subject Classification: 05C05, 05C1

Fangli Xia

Jianguang Yang

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Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 45, 2231 - 2236
The Eccentric Connectivity Index of Dendrimers
Jianguang Yang and Fangli Xia
Department of Mathematics, Hunan City University
Yiyang, Hunan 413000, P. R. China
hunancity@gmail.com
Abstract
If G is a connected graph with vertex set V , then the eccentric
connectivity index of G, ξC(G), is defined as
∑
v∈V (G)
deg(v)ecc(v) where
deg(v) is the degree of a vertex v and ecc(v) is its eccentricity. We
obtain exact formulas for calculating the eccentric connectivity index of
dendrimers.
Mathematics Subject Classification: 05C05, 05C12
Keywords: Eccentricity; Eccentric connectivity index; Dendrimers
1 Introduction
Let G be a connected graph with the vertex-set V (G) and edge-set E(G),
respectively. |V (G)| = n, |E(G)| = m are the number of vertices and edges.
The degree of a vertex v ∈ V (G) is the number of vertices joining to v and
denoted by deg(v) (or simply as dG(v), d(v)) and dG(u, v) denote the degree
of u and the distance (i.e.,the number of edges on the shortest path) between
u and v, respectively.
A critical step in pharmaceutical drug design continues to be the identifi-
cation and optimization of compounds in a rapid and cost effective way. An
important tool in this work is the prediction of physico-chemical, pharmaco-
logical and toxicological properties of a compound directly from its molecular
structure. This analysis is known as the study of the quantitative structure-
activity relationship (QSAR). In chemistry, a molecular graph represents the
topology of a molecule, by considering how the atoms are connected. This can
be modeled by a graph, where the points represent the atoms, and the edges
2232 J. Yang and F. Xia
symbolize the covalent bonds. Relevant properties of these graph models are
then studied, giving rise to numerical graph invariants. The parameters de-
rived from this graph-theoretic model of a chemical structure are being used not
only in QSAR studies pertaining to molecular design and pharmaceutical drug
design, but also in the environmental hazard assessment of chemicals. Many
such graph invariant topological indices have been studied. The first, and most
well-known parameter, the Wiener index, was introduced in the late 1940s in
an attempt to analyze the chemical properties of paraffins (alkanes)(see [1]).
This is a distance-based index, whose mathematical properties and chemical
applications have been widely researched. Numerous other indices have been
defined, and more recently, indices such as the eccentric distance sum, and
the adjacency-cum-distance-based eccentric connectivity index have been con-
sidered. These topological models have been shown to give a high degree of
predictability of pharmaceutical properties, and may provide leads for the de-
velopment of safe and potent anti-HIV compounds. Refinements of some of
these indices have also been considered.
The eccentric connectivity index of the molecular graph G, ξC(G), was
proposed by Sharma, Goswami and Madan(see [2]). It is defined as ξC(G) =
∑
v∈V (G)
deg(v)ecc(v), where ecc(v) = max{d(x, v)|x ∈ V (G)}, see [3-7] for de-
tails. The radius and diameter of G are defined as the minimum and maximum
eccentricity among vertices of G, respectively. Herein, our notation is standard
and taken from the standard book of graph theory(see[8]).
Dendrimers[10-14] are a new class of polymeric materials. They are highly
branched, mono-disperse macromolecules. The structure of these materials
has a great impact on their physical and chemical properties. As a result of
their unique behavior dendrimers are suitable for a wide range of biomedical
and industrial applications[9]. Consider the molecular graph regular dendrimer
Tk,d is a central tree with center v0 and having every non-pendent vertex is of
degree d, and the distance from v0 to each pendent vertex is k. T2,4 and T3,4
are depicted in Figure 1.
The eccentric connectivity index of dendrimers 2233
T2,4 T3,4
Figure 1. Dendrimers Tk,d for k = 2, d = 4 and k = 3, d = 4
2 Main Results
These are the main results of the paper.
For special classes of graphs we have the following useful results.
Lemma 2.1 For the complete graph Kn and bipartite graph Ka,b, we have
ξC(Kn) = n(n − 1) (for n ≥ 2) (1)
ξC(Ka,b) = n(n − 1) (for a, b 6= 1) (2)
and the index reaches its maximum for Ka,b when a = b =
n
2
.
For the star, cycle and path of order n,
Lemma 2.2 For the star Sn, cycle Cn and path Pn, we have
ξC(Sn) = 3(n − 1) (3)
ξC(Cn) =
{
n2, for n even;
n(n − 1), for n odd.
(4)
ξC(Pn) =
{
1
2
(3n2 − 6n + 4), for n even;
3
2
(n − 1)2, for n odd.
(5)
Note that the order of Tk,d is n(Tk,d) = 1 +
d
d−2
[(d − 1)k − 1].
2234 J. Yang and F. Xia
Theorem 2.3 For any pair of integers (k, d), where d ≥ 3.
ξC(Tk,d) =







dk + d
2k[1−(d−1)k−1]
2−d
+ 2dk(d − 1)k−1 + d2[1−(d−1)
k−1
(2−d)2
+k−1
d−2
(d − 1)k−1], when d ≥ 3;
6k2, when d = 2.
(6)
Proof. For the convenience of the computation, we classified the vertices
of Tk,d into k classes.
(i) For the vertices v0, the vertex for the contribution to the eccentric
connectivity index is ξC1 = dk;
(ii)For the vertices x with d(v0, x) = i, i = 1, 2, · · · , k−1: their contribution
to eccentric connectivity index are d2(d − 1)i−1(k + i), respectively, summing
up, we have
ξC2
= d2 ×
k−1
∑
i=1
(d − 1)i−1(k + i)
= d2 × { k
2−d
[1 − (d − 1)k−1] + 1−(d−1)
k−1
(2−d)2
+ k−1
d−2
(d − 1)k−1}
(iii) For the vertices x with d(v0, x) = k: their total contribution to eccen-
tric connectivity index is ξC3 = 2dk(d − 1)
k−1.
Summing up, we arrive at the desired result.
The chemically most interesting cases of equation (1) correspond to d =
3, 4:
Theorem 2.4 For the fixed k, we have
ξC(Tk,3) = 3(4k − 3) · 2
k − 3(2k − 3) (7)
ξC(Tk,4) = 4(2k − 1) · 3
k − 4(k − 1) (8)
Acknowledgments:A project supported by the Research Foundation of Ed-
ucation Bureau of Hunan Province, China(Grant NO: 10B015).
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Received: March, 2010


The Eccentric Connectivity Index of Dendrimers

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