chemia - Studia

MAHDIEH AZARI, ALI IRANMANESH, ABOLFAZL TEHRANIAN
Definition 4. Let G = ( V ( G),
E(
G))
be a graph.
(i) Let u ∈ V (G)
. Set; Δ
u
= { z ∈V
( G) :[ z,
u]
∈ E(
G)}
. In fact, Δ
u
is the
set of all vertices of G , which are adjacent to u. Suppose that, δ
u
is the number
of all vertices of G, which are adjacent to u. Clearly, δ = Δ deg( u G).
u u
=
(ii) For each pair of distinct vertices u, v ∈ V ( G)
, let δ
( u,
v)
be the
number of all vertices of G , which are adjacent both to u and v. Obviously,
δ = Δ I Δ .
( u,
v)
u
v
(iii) Let u, v and z be three vertices of G, which every pair of them is
distinct. Assume that, δ
( u,
v,
z)
denotes the number of all vertices of G which are
adjacent to vertices u, v and z. It is easy to see that, δ = Δ I Δ I Δ .
( u,
v,
z)
(iv) Suppose that, u, v and z be three vertices of graph G , which every
pair of them is distinct. Denote by N
( z, u ~ , v ~ )
, the number of all vertices of G,
which are adjacent to z, but neither to u nor to v. By the definition of N
( z, u ~ , v ~ )
,
we have:
N = Δ − ( Δ U Δ ) = Δ − Δ I ( Δ U Δ ) = Δ − ( Δ I Δ ) U ( Δ I Δ ) =
Δ
( z, u ~ , v ~ ) z u v z z u v z z u z v
z
− ( Δ
z
Δu
+ Δ
z
I Δv
− Δ
z
I Δu
I Δv
) = δ
z
− δ(
z,
u)
− δ(
z,
v)
+ δ(
z,
u,
v)
I .
Proposition 5.
( ) 1
2⎛
V G1
+
⎛ ⎞ ⎞ 1
∑d0 ( e,
f G1[
G2]
) = E( G ) ⎜
2
W ( G1
) ⎟
⎜
⎟ +
− V ( G1
) (2M1(
G2)
− N(
G2))
,
{ e,
f } ∈A
2
⎝⎝
⎠ ⎠ 4
N(
G ) =
N .
where,
2 ∑ ∑
( z , ~ , ~ 2 u2
v2
)
[ u2 , v2
] ∈E(
G2
) z2
∈V
( G2
) −(
Δ u UΔ
)
2 v2
Proof. At first, we need to find A
2
and A2 U A3
. It is easy to see that
1
1
A2
= V ( G1
) ∑ ∑N(
z , ~ , ~
2 u2
v2
)
= V ( G1 ) N(
G 2
) ,
4
4
A U A
1
2
1
2
[ u , v2
] ∈E(
G2
) z2∈V
( G2
) −(
Δ u UΔv
)
2 2 2
1
= V ( G1
) ∑
2
( E(
G
) − ( δ
+ δ
2 3
2 u2
v2
[ u2 , v2
] ∈E(
G2
)
⎛
V ( G ⎜
1)
⎝
∑
[ u v ]
E(
G
) −
∑
u
−1))
=
∑
( δ
u
+ δ ) + 1
2 v2
2
, ∈E(
G )
∈E(
G )
[ u , v ] ∈E(
G )
[ u , v ]
2 2 2
2 2 2
2 2 2
2
V ( G1 ) ( E(
G2)
+ E(
G2)
− M1(
G2))
,
v
⎞
⎟
=
⎠
z
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