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COMPUTATION OF THE FIRST EDGE WIENER INDEX OF A COMPOSITION OF GRAPHS
1
V ( G1 ) (2M1(
G2)
− N(
G2))
,
4
where Min ( G ) =
min{ d(
u , v G ), d(
u , z G )} and
N(
G ) =
2
1
∑
∑
∑
u1∈V
( G1
)[
v1
, z1
] ∈E(
G1
)
∑
N
( z , ~ , ~
2 u2
v2
)
[ u2 , v2
] ∈E(
G2
) z2∈V
( G2
) −(
Δ u UΔ
)
2 v2
Proof. Recall that, each pair of the sets A, B and C is disjoint and union of
them is the set of all two element subsets of E ( G [ G 1 2])
. Now, using the
definition of the first edge Wiener index, we obtain:
W
e
( G1[
G2])
= ∑d0(
e,
f G1[
G2])
=
+
0
∑d0 ( e,
f G1[
G2])
∑
{ e,
f } ∈B
{ e,
f } ⊆ E(
G [ G ])
1
2
.
1
1
1
{ e,
f } ∈A
1
1
1
d0 ( e,
f G1[
G2])
+ ∑d0 ( e,
f G1[
G2])
. Now, by the above Lemmas,
the proof is completed.
{ e,
f } ∈C
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