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GENERALIZED ZAGREB INDEX OF GRAPHS
Figure 9. TUVAC 5
C 7
(4,4)
Now, let T = TVAC C ( p,
) be the nanotorus related to G. Then
5 7
q
r s s r
r+
s+
1
V ( T ) = 8 pq , E( T ) = 12 pq and M ( T)
= 12 pq(3
3 + 3 3 ) = 8pq
.
3.1.5 C C C ( p,
) nanotubes and nanotori
5 6 7
q
A
5C6C7
{ r,
s}
3
C net is a trivalent decoration made by alternating C
5
, C
6
and C
7
. It can cover either a cylinder or a torus.
Let G = TUHAC C C ( p,
) (see Figure 10). We denote the number
5 6 7
q
of pentagons in the first row by p. In this nanotube, the three first rows of
vertices and edges are repeated, alternatively. We denote the number of
this repetition by q. In each period of this nanotube, there are 16p vertices
and we have q periods. So V ( G)
= 16 pq . Also, the number of edges in
each period is equal to 24p expect from the last period which has 22p
edges. So E( G)
= 24p(
q −1)
+ 22p
= 24pq
− 2 p and we have:
r s s r
r s s r
M{
r,
s}(
G)
= 8p(2
3 + 2 3 ) + (24pq
−10p)(3
3 + 3 3 ) =
r+
1 s s+
1 r
r+
s
4 p(2
3 + 2 3 + (12q
− 5)3 ) .
Figure 10. HAC C (4,2)
5 6C7
Now, let T = THAC C C ( p,
) be the nanotorus related to G. Then
5 6 7
q
V ( T ) = 16 pq , E( T ) = 24 pq and
M .
r s s r
r+
s+
1
{ r,
s}
( T)
= 24 pq(3
3 + 3 3 ) = 16 pq3
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