chemia - Studia

COMPUTATION OF THE FIRST EDGE WIENER INDEX OF A COMPOSITION OF GRAPHS
Recall that, each pair of the sets A i
( 1 ≤ i ≤ 4)
is disjoint and U 4 A = A i
,
i=
1
then by
Proposition 1, we have:
∑
∑ ∑
d0( e,
f G1[
G2])
= d0(
e,
f G1[
G2])
= A1
+ 3 A2
+ 2 A
{ e,
f } ∈ A
i= 1 { e,
f } ∈A
4
∑ { 1+
d ( u1,
v1
G1
) :{ e,
f } ∈ A4
, e = [( u1,
u2),(
u1,
v2)],
f = [( v1,
z2),(
v1,
t2)]}
=
A 3 A + A + A +
1
+
2
2
3 4
∑ { d ( u1,
v1
G1
) :{ e,
f } ∈ A4
, e = [( u1,
u2
),( u1,
v2
)], f = [( v1,
z2
),( v1,
t2
)]} =
4
U
i=
1
⎜
⎝
A
i
+
4
∑
A
2
+ ( A2
+ A3
) + A2
+ E(
G2
) ∑ d(
u1,
v
i
i= ⊆V
( G )
i
{ u , v }
1 1 1 1
A U A
2
⎛ V ( G1 ) E(
G 2 )
2
2
⎞
⎟
⎠
3
+
E ( G2)
W ( G1
) =
1
2
V ( G ) E(
G
2
A
2
2
1
G ) =
+ E( G2
) W ( G1
) = A + A2
U A3
+ A2
+ E(
G2
) W ( G1
) =
1
2
1
+ V ( G1 ) ( E(
G2)
+ E(
G2)
− M1(
G2))
+ V ( G1 ) N(
G 2
) +
2
4
2
2
( ) − V ( G ) E(
G ) + V ( G ) E(
G ) + V ( G ) E(
) ) −
1 2
1
2
1
2
1
G2
1
1
2
V ( G1 ) M1(
G2)
+ V ( G1 ) N(
G 2
) + E ( G2)
W ( G1
) =
2
4
( ) 1
2⎛
V G1
+
⎛ ⎞ ⎞ 1
E( G ) ⎜
2
W ( G1
) ⎟
⎜
⎟ +
− V ( G1
) (2M1(
G2)
− N(
G2))
2
⎝⎝
⎠ ⎠ 4
Proposition 6.
V ( G )
2⎛
2
⎞
4
∑d0 ( e,
f G1[
G2]
) = V ( G2)
⎜ ⎟M1(
G1
) + V ( G2)
We
( G1
)
0
{ e,
f } ∈B
⎝ 2 ⎠
Proof. For the proof of this proposition, we need to obtain B , B and
1 2
B
4
. It is easy to see that:
1
2
3
+
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