Computing sanskrti index of V-PHENYLENIC NANOTUBES AND NANTORIV-PHENYLENIC NANOTUBES AND NANTORI [355]

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International Journal of Pure and Applied Mathematics
Volume 115 No. 4 2017, 859-865
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: 10.12732/ijpam.v115i4.19
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A
ijpam.eu
COMPUTING SANSKRUTI INDEX OF
V-PHENYLENIC NANOTUBES AND NANOTORI
H. Jiang1, M.S. Sardar2, M.R. Farahani3§,
M. Rezaei4, M.K. Siddiqui5
1School of Information Science and Engineering
Chengdu University
Chengdu, 610106, P.R. CHINA
2University of Management and Technology (UMT)
Lahore, PAKISTAN
3Department of Applied Mathematics
Iran University of Science and Technology (IUST)
Narmak, Tehran, 16844, IRAN
4Department of Mathematics
Buein Zahra Technical University
Buein Zahra, Qazvin, IRAN
Department of Mathematics
5Comsats Institute of Information Technology
Sahiwal, PAKISTAN
Abstract: Among topological descriptors connectivity topological indices are very important
and they have a prominent role in chemistry. One of them is Sanskruti index defined as
S(G) = Puv∈E(G)(SuSv
Su+Sv−2)3where Suis the summation of degrees of all neighbors of vertex
uin G. In this paper we compute this topological index for V-phenylenic nanotube and
nanotori.
AMS Subject Classification: 05C90, 05C35, 05C12
Key Words: molecular graph, benzenoid, Capra operation, topological index, Sanskruti
index
Received: May 4, 2017
Revised: July 10, 2017
Published: August 9, 2017
c
2017 Academic Publications, Ltd.
url: www.acadpubl.eu
§Correspondence author

860 H. Jiang, M.S. Sardar, M.R. Farahani, M. Rezaei, M.K. Siddiqui
1. Introduction and Preliminaries
Let G= (V;E) be a simple molecular graph without directed and multiple
edges and without loops, the vertex and edge sets of it are represented by
V=V(G) and E=E(G), respectively. In chemical graphs, the vertices
correspond to the atoms of the molecule, and the edges represent to the chemical
bonds. Also, if eis an edge of G, connecting the vertices uand v, then we write
e=uv and say uand vare adjacent.
Mathematical chemistry is a branch of theoretical chemistry for discussion
and prediction of the molecular structure using mathematical methods without
necessarily referring to quantum mechanics. Chemical graph theory is a branch
of mathematical chemistry which applies graph theory to mathematical mod-
eling of chemical phenomena ([1]-[5]). This theory had an important effect on
the development of the chemical sciences.
Among topological descriptors, connectivity indices are very important and
they have a prominent role in chemistry. In other words, if Gbe the connected
graph, then we can introduce many connectivity topological indices for it, by
distinct and different definition. A connected graph is a graph such that there
is a path between all pairs of vertices. One of the best known and widely used is
the connectivity index, introduced in 1975 by Milan Randi´c [6], who has shown
this index to reflect molecular branching and defined as follows:
R(G) = X
uv∈E(G)
1
√dudv
The Sanskruti index S(G) of a graph Gis defined as follows (see [7]-[11]):
S(G) = X
uv∈E(G)
(SuSv
Su+Sv−2)3.
where Suis the summation of degrees of all neighbors of vertex uin G. In Refs
[12]-[27] some topological indices of V−phenylenic nanotube and V−phenylenic
nanotori are computed. In this paper, we continue this work to compute the
Sanskruti index of molecular graphs related to V−phenylenic nanotube and
nanotori. Our notation is standard and mainly taken from Refs. [1]-[5].
2. Main Results and Discussion
The goal of this section is to computing the Sanskruti index of V−phenylenic
nanotube and nanotori. The novel phenylenic and naphthylenic lattices pro-

COMPUTING SANSKRUTI INDEX OF... 861
posed can be constructed from a square net embedded on the toroidal surface.
Phenylenes are polycyclic conjugated molecules, composed of four membered
ring (=square) and six-membered rings (=hexagons) such that every four mem-
bered ring (4-membered cycle) is adjacent to two 6-membered cycles, and no
two six-membered rings are mutually adjacent. Each four-membered ring lies
between two six-membered rings, and each hexagon is adjacent only two four-
membered rings. Because of such structural features phenylenes are very inter-
esting conjugated species [28]-[33]. The rapid development of the experimental
study of phenylenes motivated a number of recent theoretical studies of thee
conjugated π-electron systems [33].
Following M. V. Diudea [5] we denote a V-Phenylenic nanotube and V-
Phenylenic nanotorus by G=V P H X[m, n] and H=V P HY [m, n], respec-
tively. The general representation of these nano structures are shown in Figure
1 and Figure 2. For more information and background materials, refer to paper
series [12]-[33] again. Now we have following theorems, immediately.
Theorem 2.1. ∀m, n ∈N, the Sanskruti index S(G)of V−Phenylenic
Nanotube V P HX [m, n]is equal to
S(V P HX [m, n]) = −87714218531
175616000 m+4782969
4096 mn.
Figure 1: The Molecular Graph of V−Phenylenic Nanotube
V P H X [m, n].

862 H. Jiang, M.S. Sardar, M.R. Farahani, M. Rezaei, M.K. Siddiqui
Proof. Consider the V−phenylenic nanotube G=V P H X [m, n] with 6mn
vertices and 9mn −medges (Figure 1). In V−phenylenic molecule, there are
two partitions V2=v∈V(G)|dv= 2 and V3=v∈V(G)|dv= 3 of
V(V P HX [m, n]), since the degree of an arbitrary vertex/ atom of a molec-
ular graph is equal to 2 or 3. Next, the two partitions of E(G) are E5={u, v ∈
V(G)|du= 3&dv= 2}and E6={u, v ∈V(G)|du=dv= 3}.
Also, two adjacent vertices v1,v2of a vertex v∈V2 have degree three, then
Sv= 2×3 = 6 and two edges vv1and vv2belong to E5(and |E5|= 2|V2|= 4m).
Also, for all vertices uin first and end row of V−phenylenic nanotube with
degree three, N(u) = v1, v2, v3such that v1∈V2and v2, v3∈V3(uv1∈E5and
uv3, uv2∈E6), thus Su= 2 ×3 + 2 = 8.Finally, for other vertices Sw= 9,
because all other vertices and their edges belong to V3and E6, respectively. So,
the Sanskruti index S(G) of V P H X[m, n](m, n ≥1) will be
S(V P H X [m, n] = X
uv∈E(G)
(SuSv
Su+Sv−2)3
=4m.(6.8
6 + 8 −2)3+ 2m.(8.8
8 + 8 −2)3+ 2m.(8.9
8 + 9 −2)3
+ (9mn −9m).(9.9
9 + 9 −2)3
=−87714218531
175616000 m+4782969
4096 mn.
Theorem 2.2. ∀m, n ∈N, the Sanskruti index S(G)of V−Phenylenic
Nanotori H=V P H Y [m, n]is equal to
S(V P HY [m, n]) = 4782969
4096 mn.
Proof. The proof is easily, since by considering the V−phenylenic nan-
otori H=V P H Y [m, n] with 6mn vertices and 9mn edges (Figure 2). We
see that this nanotori is a Cubic graph and all vertices belong to V3and
∀v∈V(V P HY [m, n]) Sv= 9. This implies that all edges belong
S(V P H X [m, n] = X
uv∈E(G)
(SuSv
Su+Sv−2)3
=(9mn).(9.9
9 + 9 −2)3=4782969
4096 mn.

COMPUTING SANSKRUTI INDEX OF... 863
Figure 2: The Molecular Graph of V-Phenylenic Nanotorus
V P H Y [m, n].
3. Conclusions
In this report, we study some properties of a new connectivity index of (molec-
ular) graphs that called Sanskruti index. This connectivity index was defined
as follows:
S(G) = X
uv∈E(G)
(SuSv
Su+Sv−2)3.
where Suis the summation of degrees of all neighbors of vertex uin G. In
continue, closed analytical formulas for S(G) of a physico chemical structure
of phenylenic nanotubes and nanotorus are given. These nano structures are
V−Phenylenic Nanotube V P HX [m, n] and V-phenylenic nanotorus V P H Y [m,
n].
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