Chemistry Reference
In-Depth Information
Chapter 7
The First and Second Zagreb Indices
of Several Interesting Classes of Chemical
Graphs and Nanostructures
Ali Iranmanesh and Mahdieh Azari
Abstract
Zagreb indices are among the oldest topological indices, and were intro-
duced by Gutman and Trinajstic in 1972. In this chapter, we study the behavior of
the first and second Zagreb indices under some graph decorations such as the bridge
graphs, chain graphs and rooted product of graphs. Zagreb indices of the cluster
of graphs and thorn graphs as two important special cases of rooted product are
also determined. Using these results, the first and second Zagreb indices of several
interesting classes of chemical graphs and nanostructures will be computed.
7.1
Introduction
In this chapter, we consider connected finite graphs without any loops or multiple
edges. Let G be such a graph with the vertex set
V
(
G
) and the edge set
E
(
G
). For
u
V
(
G
), we denote by
N
G
(
u
) the set of all neighbors of
u
in
G
. Cardinality of
the set
N
G
(
u
) is called the degree of u in
G
and will be denoted by deg
G
(
u
). We
denote by
α
G
(
u
), the sum of degrees of all neighbors of the vertex
u
in
G
, i.e.,
α
G
(
u
)
∈
=
a
∈
N
G
(
u
)
deg
G
(
a
). We denote by
|S|
the cardinality of a set
S
.
In theoretical chemistry, the physico-chemical properties of chemical compounds
are often modeled by the molecular- graph-based structure-descriptors which are
also referred to as topological indices (Gutman and Polansky
1986
; Trinajstic
1992
;
Todeschini and Consonni
2000
). The Zagreb indices are among the oldest topological
indices, and were introduced as early as in 1972 (Gutman and Trinajstic
1972
).
For details on their theory and applications see (Gutman et al.
1975
; Nikoli´cetal.
2003
; Zhou
2004
; Das and Gutman
2004
; Azari and Iranmanesh
2011a
; Réti
2012
;
Stevanovic and Milanic
2012
; Eliasi et al.
2012
).
