chemia - Studia

MAHDIEH AZARI, ALI IRANMANESH, ABOLFAZL TEHRANIAN
∑
d0 ( e,
f G1[
G2]
) =
{ e,
f }∈ C 3
{ 1+
min{ d ( u1,
v1
G1
), d(
u1,
z1
G1
) :{ e,
f } ∈C3,
e = [( u1,
u2),(
u1,
t2)],
f = [( v1,
v2),(
z1,
z2
2
C3 + E(
G2)
V ( G2)
∑ ∑ min{ d(
u1,
v1
G ), d(
u1,
z1
G1
)} =
1
u1∈
V ( G1
) [ v1
, z1
] ∈E
( G1
)
v1
≠z1
, z≠1u1
2
C
3
+ E(
G2)
V ( G2)
∑ ∑ min{ d(
u1,
v1
G ), d(
u1,
z1
G1
)} =
1
u1∈ V ( G1
)[
v1
, z1
] ∈E(
G1
)
2
C
3
+ E( G2)
V ( G2)
Min(
G1
).
∑ )]} =
Since each pair of the sets C i
( 1 ≤ i ≤ 3)
is disjoint and U 3 C =
i=
1
d ( e,
f G [ G ] = d ( e,
f G [ G ] + d ( e,
f G [ G
∑
194
∑
C i
, we have:
0 1 2
)
0 1 2
)
0 1 2
])
=
{ e,
f } ∈C
{ e,
f } ∈C1
UC2
{ e,
f }∈ C 3
2
C
1
+ 2C2
+ C3
+ E(
G2
) V ( G2
) Min(
G1
) =
3
2
∑ Ci
+ C2
+ E( G2
) V ( G2
) Min(
G1
) =
i=
1
3
2
U Ci
+ C2
+ E( G2)
V ( G2)
Min(
G1
) =
i=
1
2
2
C + C2 + E( G2)
V ( G2)
Min(
G1
) = V ( G1 ) V ( G2
) E(
G1
) E(
G2
) +
2
2 E ( G1 ) E(
G2
) V ( G2
) ( V ( G2
) − 2)
+ E ( G2 ) V ( G2
) Min(
G1
) =
2
( G1 ) E(
G2)
V ( G2)
( V ( G1
) V ( G2)
+ 2V
( G2)
− 4) + E( G2)
V ( G2)
Min(
G1
E )
Now, as the main purpose of this paper, we express the following theorem, which
characterizes the first edge Wiener index of the composition of two graphs.
Theorem. Let G
1
= ( V ( G1
), E(
G1
)) and G
2
= ( V ( G2
), E(
G2
)) be two
simple undirected connected finite graphs, then
V ( G
⎛
1)
+1
2 ⎞
W e
( G1[
G2])
= E(
G +
0 2)
⎜
⎟
⎝ 2 ⎠
E( G1 ) E(
G2)
V ( G2)
( V ( G1
) V ( G2)
+ 2V
( G2)
− 4)
+
V ( G
2
4
⎛
2 )
2 ⎞
E( G2)
W ( G1
) + V ( G2)
We ( G1
) + V ( G )
+
0 ⎜
⎟
2
M1(
G1
)
⎝ 2 ⎠
E ( G2)
V ( G2)
Min(
G1
) −
2
∑