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Copyright: American Scientific Publishers
RESEARCH ARTICLE
Copyright © 2015 American Scientific Publishers
All rights reserved
Printed in the United States of America
Journal of
Computational and Theoretical Nanoscience
Vol. 12, 3860–3863, 2015
The Adjacent Eccentric Distance Sum Index of
One Pentagonal Carbon Nanocones
Fereshteh Momen and Mehdi Alaeiyan
∗
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, 31485-313, Iran
Let G = V G EG be a simple connected graph. The adjacent eccentric distance sum index
of G is defined as AEDSG =
∑
u∈V G
ecc uDu/degu, where ecc u is the largest distance
between u and any vertex v of graph G and degu is the degree of u and Du =
∑
v ∈V G
duv
is the sum of all distances from the vertex u. In this paper we present an exact formula for the
adjacent eccentric distance sum index of one pentagonal carbon nanocones.
Keywords: Carbon Nanocones, Distance Sum, Eccentricity, Molecular Graph, Topological Index,
The Adjacent Eccentric Distance Sum Index.
1. INTRODUCTION
A molecular graph is a simple graph such that its vertices
correspond to the atoms and the edges to the bonds. A path
of a length n in a graph G is a sequence of n + 1 vertices
such that from each of these vertices there is an edge to the
next vertex in the sequence. A vertex is external, if it lies
on the boundary of the unbounded face of G, otherwise,
the vertex is called internal.
Let G be molecular graph, with the vertex and edge sets
of which are represented by V G and EG, respectively.
The distance du v is defined as the length of the short-
est path between u and v in G. Du denotes the sum
of distances between u and all other vertices of G. For
a given vertex u of V G its eccentricity, eccu, is the
largest distance between u and any other vertex v of G.
The maximum eccentricity over all vertices of G is called
the diameter of G and denoted by DiamG and the mini-
mum eccentricity among the vertices of G is called radius
of G and denoted by RG.
Research into carbon nanocones (CNC ) started almost at
the same time as the discovery of carbon nanotubes CNT
in 1991. Ball studied the closure of CNT and mentioned
that (CNT ) could sealed by a conical cap.
1
The official
report of the discovery of isolated CNC was made in
1994, when Ge and Sattler reported their observations of
carbon nanocones mixed together with tubules and a flat
graphite surface.
2
These are constructed from a graphene
sheet by removing a 60
wedge and joining the edges pro-
duces a cone with a single pentagonal defect at the apex.
∗
Author to whom correspondence should be addressed.
Topological indices are graph invariants and are used
for Quantitives Structure-Activity Relationship (QSAR)
and Quantitives Structure-Property Relationship (QSPR)
studies.
3 4
Many topological indices have been defined
and several of them have found applications as means
to model physical, chemical, pharmaceutical and other
properties of molecules. Several topological indices are
based on graph theoretical notion of eccentricity have been
recently proposed. Namely, eccentric connectivity index
5 6
and augmented and superaugmented eccentric connectiv-
ity index.
7–11
These indices have been shown to be very
useful, therefore their mathematical properties have been
studied too.
Recently a novel graph invariant for predicting bio-
logical and physical properties, eccentric distance sum
EDS was introduced by Gupta et al.,
12
which was defined
as EDSG =
∑
u∈V G
eccuDu. For more details, see
Ref. [13].
The Wiener index is one of the most used topo-
logical indices with high correlation with many physi-
cal and chemical indices of molecular compounds. The
Wiener index of a graph G denoted by W G is defined,
W G = 1/2
∑
u∈V G
Du. The parameter DDG is
called the degree distance of G and it was introduced
by Dobrynin and Kochetova
14
and Gutman
4
as a graph-
theoretical descriptor for characterizing alkanes, it can be
considered as a weighted version of the Wiener index, is
defined as, DDG =
∑
u∈V G
deguDu. When G is a
tree on n vertices, it has been demonstrated the Wiener
index and degree distance are closely related by DDG =
4W G - nn - 1.
3860 J. Comput. Theor. Nanosci. 2015, Vol. 12, No. 10 1546-1955/2015/12/3860/004 doi:10.1166/jctn.2015.4293