chemia - Studia

MAHDIEH AZARI, ALI IRANMANESH, ABOLFAZL TEHRANIAN
The Zagreb indices have been defined more than thirty years ago
by Gutman and Trinajestic, [14].
Definition 1. [14] The first Zagreb index of G is defined as:
2
M ( G)
= deg( u G .
∑
1
)
u∈V
( G)
The edge versions of Wiener index of G , which were based on the
distance between all pairs of edges of G , were introduced by Iranmanesh et al.
in 2009 [15]. We encourage the reader to consult [16-20], for computational
techniques and mathematical properties of the edge Wiener indices. The
first edge Wiener index of G , is defined as follows:
Definition 2. [15] The first edge Wiener index of G , is denoted by W e
( G)
.
0
That is:
W
e
( G)
= ∑ d
0
( e,
f G)
, where
⎪⎧
d1(
e,
f G)
+ 1 if e ≠ f
d ( e,
f G)
=
0
⎨
and
0
{ e,
f } ⊆E(
G)
⎪⎩ 0
if e = f
( e,
f G)
min{ d(
u,
z G),
d(
u,
t G),
d(
v,
z G),
d(
v,
t )} e = u,
v ,
d = , such that [ ]
1
G
f = [ z,
t]
. This index satisfies the relation We ( G)
= W ( L(
G))
0 v
, where L(G)
is
the line graph of G.
In this paper, we want to find the first edge Wiener index of the
composition of graphs.
Recall definition of the composition of two graphs.
Definition 3. Let G
1
= ( V ( G1
), E(
G1
)) and G
2
= ( V ( G2
), E(
G2
)) be two
connected graphs. We denote the composition of G 1
and G
2
by G [ G 1 2
] ,
that is a graph with the vertex set V ( G1[
G2
]) = V ( G1
) × V ( G2
) and two
vertices u , ) and v , ) of G ] are adjacent if and only if:
( u ( v
1 2
1 2
1
v1
and [ u2
, v2
] ∈ E(
G2
[ G 1 2
[ u = ) ] or [ u1,
v1]
∈ E(
G1
) .
By definition of the composition, the distance between every pair of
distinct vertices u = u 1,
u ) and v = v , v ) 1 of G ] , is equal to
d ( u,
v G [ G
1
2
(
2
⎧d(
u1,
v
⎪
]) = ⎨1
⎪
⎪2
⎩
1
G )
1
if
if
if
u
1
1
( 2
1
≠ v
[ G 1 2
Let G
1
= ( V ( G1
), E(
G1
)) and G
2
= ( V ( G2
), E(
G2
)) be two graphs.
Consider the sets E
1
and E
2
as follows:
184
1
u = v , [ u
1
u = v , v
1
2
2
, v ] ∈ E(
G )
2
2
is not adjacent to u
2
in G
COMPUTATION OF THE FIRST EDGE WIENER INDEX OF THE COMPOSITION
OF GRAPHS
2