chemia - Studia

COMPUTATION OF THE FIRST EDGE WIENER INDEX OF A COMPOSITION OF GRAPHS
E1 = {[( u1,
u2),(
u1,
v2)]
∈ E(
G1[
G2]) : u1
∈V
( G1
), [ u2,
v2]
∈ E(
G2)}
E2 = {[( u1,
u2),(
v1,
v2)]
∈ E(
G1[
G2]) :[ u1,
v1
] ∈ E(
G1
), u2,
v2
∈V
( G2)}
By definition of the composition, E U E = E G [ ]) and obviously,
E I = φ , E = V G ) E(
) and E = V G ) E(
) .
1
E 2
1 2
(
1
G2
2
1
(
1
G2
2
(
2
G1
Set:
A = {{ e,
f } ⊆ E(
G1[
G2]) : e ≠ f , e,
f ∈ E1}
B = {{ e,
f } ⊆ E(
G1[
G2]) : e ≠ f , e,
f ∈ E2}
C = {{ e,
f } ⊆ E(
G1[
G2]) : e∈
E1,
f ∈ E2}
It is easy to see that each pair of the above sets is disjoint and the
union of them is the set of all two element subsets of E G 1[
G ]) . Also we have:
E
⎛
1 ⎛ V ( G
⎞
1)
E(
G2
) ⎞
A = = ⎜ ⎟
⎜
⎟
,
⎝ 2 ⎠
2
⎝ ⎠
2
E
⎛
2 ⎛ V ( G
⎞
2 ) E(
G1
) ⎞
B = = ⎜ ⎟
⎜
⎟
,
⎝ 2 ⎠
2
⎝ ⎠
(
2
C = E1 E2
= V ( G1
) V ( G2
) E(
G1
) E(
G2
)
2
Consider four subsets A
1
, A2
, A3
and A
4
of the set A as follows:
A1 = {{ e,
f } ∈ A : e = [( u1,
u2),(
u1,
v2)],
f = [( u1,
u2),(
u1,
z2)],
u1
∈V
( G1
),
u v , z ∈ V ( )}
A
2, 2 2
G2
= {{ e,
f } ∈ A : e = [( u1,
u2),(
u1,
v2)],
f = [( u1,
z2),(
u1,
t2)],
u1
∈V
(
1),
u , v , z , t ∈V ( G ), both z and t are adjacent neither
2
G
2 2 2 2 2 2 2
to u nor to v in G }
2 2 2
A3 = {{ e,
f } ∈ A : e = [( u1,
u2),(
u1,
v2)],
f = [( u1,
z2),(
u1,
t2)],
u1
∈V
( G1
),
u2 , v2
∈ V ( G2),
z2,
t2
∈V
( G2)
−{
u2,
v2}}
− A2
A4 = {{ e,
f } ∈ A : e = [( u1,
u2),(
u1,
v2)],
f = [( v1,
z2),(
v1,
t2)],
u1,
v1
∈V
( G1
),
v ≠ u u , v , z , t ∈V
( )}
1 1, 2 2 2 2
G2
It is clear that, every pair of the above sets is disjoint and U 4 A = A i
.
i=
1
In the next Proposition, we characterize d e,
f G [ ]) for all { e,
f } ∈ A .
Proposition 1. Let { e,
f } ∈ A .
0( 1
G2
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