chemia - Studia

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2
1
2
1
2
1
2
B
COMPUTATION OF THE FIRST EDGE WIENER INDEX OF A COMPOSITION OF GRAPHS
4
3
( V ( G2)
− V ( G2)
)
V ( G
V ( G
∑
∑
∑
[ u1
, v1
] ∈E
( G1
) z1∈{ u1
, v1
},
[ z , t ] ∈E
( G
∑
1 1
d ([ u , v ],[ z , t ] G ) + B
0
1)
4
2
)
d0([
u1,
v1],[
z1,
t1]
G1
)
[ u1
, v1
] ∈E
( G1
) [ z1
,
] ∈E
( G1
),
z1
, t1∉{ u1
, v1
}
2
V ( G
)
4
∑
∑
[ u1
, v1
] ∈E(
G1
) z1∈{ u1
, v1
},
[ z , t ] ∈E(
G
∑
1 1
d ([ u
∑
0
1 )
{ u , v }
1
, v
1
],[ z
1
1
, t
1
1
1
=
1
1
] G ) + B
4
2
)
d
0
([ u1,
v1],[
z1,
t1]
G1
)
[ u1
, v1
] ∈E(
G1
) [ z1
, t1
] ∈E
( G1
),
z , t ∉
1 1
1
1
4
4
+ V ( G2)
(2W
e
( G1
)) = B4
+ V ( G2)
W ( G1
)
0 e
.
2
4 0
1
Now, since U 5 B = B i
, we have:
i=
1
d ( e,
f G [ G ] =
d ( e,
f G [ G ] +
d ( e,
f G [ G ] +
∑
0 1 2
)
{ e,
f } ∈B
∑ d
0
( e,
f G1[
G2
])
{ e,
f } ∈B3
UB4
UB5
=
∑
0 1 2
)
{ e,
f }∈B
1
=
4
1
+
4
+
∑
0 1 2
)
{ e,
f }∈ B 2
V ( G
⎛
2 )
⎛
V ( G
⎞ ⎛
2 )
4
⎞
2 ⎞
B + + +
=
⎜
⎟
⎜
⎟
+
⎜
⎟
1
2 B2
B4
V ( G2)
We
( G1
) 2 E(
G1
) V ( G2)
2 − V ( G2)
+
0
⎝ 2 ⎠ ⎝ ⎝ 2 ⎠ ⎠
V ( G )
V ( G )
⎛
2
2 ⎞
4
2⎛
2
⎞
4
V ( G )
⎜
⎟
2
M1(
G1
) + V ( G2)
We
( G1
) = V ( G ) ( ) ( ) ( )
0
2 ⎜
⎟M1
G1
+ V G2
We
G .
0 1
⎝ 2 ⎠
⎝ 2 ⎠
Proposition 7.
d ( e,
f G [ G
∑
0 1 2]
) =
{ e,
f } ∈C
2
E ( G1)
E(
G2
) V ( G2
) ( V ( G1
) V ( G2
) + 2V
( G2
) − 4) + E(
G2
) V ( G2
) Min(
G1
where,
Min ( G ) =
1
∑
∑
u1∈V
( G1
)[
v1
, z1
] ∈E(
G1
)
min{ d(
u , v
Proof. First, we find C 2
and ∑d
1
1
G ), d(
u , z
0
( e,
f G1[
G2]
)
{ e,
f }∈C
3
1
1
1
G )}
1
. It is easy to see
that:
C2
= V ( G2)
( V ( G2)
− 2) E(
G2)
∑δ
u
= 2 E(
G1
) E(
G2)
V ( G2)
( V ( G2)
− 2)
1
and by Proposition 3, we have:
u1∈V
( G1
)
)
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