Chemistry Reference
In-Depth Information
The first and second Zagreb indices of a graph
G
are denoted by
M
1
(
G
) and
M
2
(
G
), respectively and are defined as follows:
deg
G
(
u
)
2
M
1
(
G
)
=
and
M
2
(
G
)
=
deg
G
(
u
)deg
G
(
v
)
.
u
∈
V
(
G
)
uv
∈
E
(
G
)
The first Zagreb index can also be expressed as a sum over edges of
G
:
M
1
(
G
)
=
[deg
G
(
u
)
+
deg
G
(
v
)]
.
uv
∈
E
(
G
)
Let
P
n
,
C
n
, and
S
n
denote the path, cycle and star on
n
vertices. It is easy to that;
M
1
(
P
n
)
=
4
n
−
6,
(
n
≥
2),
M
2
(
P
2
)
=
1,
M
2
(
P
n
)
=
4
n
−
8,
(
n
≥
3),
M
1
(
C
n
)
=
M
2
(
C
n
)
=
4
n
,
(
n
≥
3),
1)
2
,(
n
M
1
(
S
n
)
=
n
(
n
−
1),
M
2
(
S
n
)
=
(
n
−
≥
2)
.
We denote by
K
1
, the single vertex graph and assume that
P
1
=
S
1
=
K
1
. Clearly,
M
1
(
K
1
)
0.
The fact that many interesting graphs are composed of simpler graphs that serve
as their basic building blocks, prompted interest in the type of relationship that
exists between topological indices of graph operations and their building blocks. For
example see (Yeh and Gutman
1994
; Sagan et al.
1996
; Yan et al.
2007
; Došlic
2008
;
Azari and Iranmanesh
2011b
; Azari et al.
2010
,
2012
; Eliasi and Iranmanesh
2013
).
The corresponding relationships between Zagreb indices of some graph operations
and their components were established in (Khalifeh et al.
2009
; Arezoomand and
Taeri
2013
). In this chapter, this development is followed for Zagreb indices of
some other classes of composite graphs. The chapter is organized as follows. In
Sects. 7.2 and 7.3, we compute the first and second Zagreb indices for two kinds
of bridge graphs. In Sect. 7.4, we present explicit formulae for the first and second
Zagreb indices of the chain graphs. In Sect. 7.5, we study the behavior of the first
and second Zagreb indices under the rooted product of graphs. At the end of each
section, we also present some applications of our results in computing Zagreb indices
of some chemical graphs and nano-structures.
=
M
2
(
K
1
)
=
7.2
Zagreb Indices of the Bridge Graph B
1
In this section, we compute the first and second Zagreb indices for the bridge graph
B
1
. All of the results of this section have been reported in (Azari et al.
2013
). We
start this section by definition of this class of composite graphs.
Let
i
=
1
be a set of finite pairwise disjoint graphs with distinct vertices
v
i
∈
V
(
G
i
). The bridge graph
B
1
=
B
1
(
G
1
,
G
2
,
...
,
G
d
;
v
1
,
v
2
,
...
,
v
d
)of
{
G
i
}
i
=
1
{
G
i
}
with
