Degree-based indices computation for special chemical molecular and Nanotubes [410]

Abstract

Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. Suppose G is a connected molecular graph and vertices u,vεV(G). In this paper, we present explicit formulas for calculating the “general harmonic index, harmonic index and Harmonic polynomial” of a special chemical molecular graph “Cas(C)-CaR(C)[m,n,p] Nanotubes Junction” are given. The Cas(C)-CaR(C)[m,n,p] Nanotubes Junction is a new nano-structure that was defined by M.V. Diudea, on based the new graph operations (Leapfrog Le and Capra Ca) on the cycle graph Cn. In this paper, we compute the harmonic index vie two ways namely degree-based method and via polynomial method.
Keywords: Molecular graphs, Carbon Nanocones, Cas(C)-CaR(C)[m,n,p] Nanotubes Junction, Harmonic index, Harmonic polynomial, General harmonic index

Full text

International Journal of Chemical Modeling ISSN: 1941-3955
Volume 9, Number 2-3 © 2018 Nova Science Publishers, Inc.
DEGREE-BASED INDICES COMPUTATION FOR
SPECIAL CHEMICAL MOLECULAR AND NANOTUBES
Wei Gao1,

, Mohammad Reza Farahani2,
†
and Jia-Bao Liu3,
‡
1School of Information Science and Technology,
Yunnan Normal University, Kunming, China
2Department of Applied Mathematics of Iran University of Science and Technology
(IUST), Narmak, Tehran, Iran
3School of Mathematics and Physics,
Anhui Jianzhu University, Hefei, P. R. China
ABSTRACT
In this paper, formulas for calculating the Several topological indices “general Randić
index, Randić connectivity index, Sum-connectivity Index, general sum connectivity
index, first and second Zagreb indices, first and second Hyper Zagreb indices, first and
second Zagreb polynomials, general harmonic index, harmonic index” for a special
chemical molecular graph “Cas(C)-CaR(C)[m,n,p] Nanotubes Junction” are given. The
Cas(C)-CaR(C)[m,n,p] Nanotubes Junction is a new nano-structure that was defined by
M.V. Diudea, on based the new graph operations (Leapfrog Le and Capra Ca) on the
cycle graph Cn.
Keywords: molecular graph, nanotubes, general Randić index, Randić connectivity index,
Sum-connectivity Index, general sum connectivity index, first and second Zagreb indices,
first and second Hyper Zagreb indices, first and second Zagreb polynomials, general
harmonic index, harmonic index
1. INTRODUCTION
A graph is a collection of points and lines connecting a subset of them. The points and
lines of a graph also called vertices and edges of the graph, respectively. The vertices and
edges of a graph also correspond to the atoms and bonds of the molecular graph, respectively.
If e is an edge/bond of G, connecting the vertices/atoms u and v, then we write e = uv and say
“u and v are adjacent.”
 Corresponding Email: Gaowei@ynnu.edu.cn.
†† Email: MrFarahani88@Gmail.com, Mr_Farahani@Mathdep.iust.ac.ir.
‡‡ Email: liujiabaoad@163.com.

International Journal of Chemical Modeling ISSN: 1941-3955
Volume 9, Number 2-3 © 2018 Nova Science Publishers, Inc.
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 an important branch of graph theory, which applies graph
theory to mathematical modeling of chemical phenomena [1-4].
Topological indices, as numerical parameters of molecular structures, play a vital role in
chemistry, and medicine science. It has been proved that topological indices reflect
biochemical properties (such as the melting point, boiling point, toxicity and QSPR/QSAR
study) of their corresponding compounds and drugs. Several articles contributed to
determining the topological indices of special molecular graphs. There are many indices for a
connected graph G.
The Wiener index is the first reported distance based topological index and is defined as
half sum of the distances between all the pairs of vertices in a molecular graph and introduced
by chemist Harold Wiener in 1947 [5]. The Wiener index is equal to [6-9]:
where the distance dG(u,v) or d(u,v) between u,v

V (G) is defined as the length of any
shortest path in G connecting u and v.
In mathematical chemistry there are many connectivity topological indices for connected
structure graphs. The Randić Connectivity Index χ(G) is oldest of them, where introduced by
Milan Randić in 1975 [10], who has shown this index to reflect molecular branching and
defined as:
where du and dv are the degrees of the vertices u and v, respectively.
Let G be a (molecular) graph with vertex and edge sets being denoted by V(G) and E(G),
respectively. Bollobas and Erdos [11] defined the general Randić index which was stated as:
where k is a real number and dv denotes the degree of vertex v in G. Li and Liu [12] proposed
the first three minimum general Randić indices of tree structure, and they also determined the
corresponding extremal trees. Liu and Gutman [13] characterized several estimating on
general Randić index. Throughout our paper, we always assume that k is a real number.
In 2009, a closely related variant of the Randić connectivity index called the Sum-
connectivity Index was introduced by B. Zhou and N. Trinajstić [14]. For a connected graph
   
( ) ( )
1,
2
v V G u V G
vG d uW
� �

� �
 
( )
1
e uv E G u v
Gd d

�

�
   
 
k
k u v
e uv E G
R G d d
�

�

Degree-Based Indices Computation for Special Chemical Molecular and Nanotubes
G, its sum-connectivity index X(G) is defined as the sum over all edges of the graph of the
terms that
B. Zhou and N. Trinajstić [15] introduced the general sum connectivity index as
By setting k = 1 of the general Randić index and k = 1 and k = 2, respectively, then it
becomes the second Zagreb index M2(G), the first Zagreb index M1(G) and the first Hyper-
Zagreb index HM1(G):
Here, we define a new version of Zagreb topological indices, based on the Hyper-Zagreb
index that defined as above. Then, the Second Hyper-Zagreb index of a graph G, which is
defined as the sum of the weights (dudv)2 and is equal to
The first Zagreb index was formally introduced by I. Gutman and N. Trinajstić [16-18]
on based structure descriptor about forty years ago (in 1972) as the sum of the squares of the
degrees of all vertices/atoms in the molecules G, in terms of bonds and the second Zagreb
index M2(G) was conceived somewhat later [19].
Note that a new version of Zagreb indices named Hyper-Zagreb index was introduced by
Shirdel et al. [20].
Several polynomials related to degree based indices are also introduced. For instance, the
first and the second Zagreb polynomials are expressed by
1
u v
d d
 
( )
1
e uv E G u v
X G d d
�


�
   
( )
k
k u v
uv E G
X G d d
�
 
�
 
1
( )
( )
u v
e uv E G
M G d d
�
 
�
 
2
( )
( )
u v
e uv E G
M G d d
�
�
�
   
 
2
v
v E G
v
e u
HM G d d
�
 �
   
 
2
2
e v E G
v
u
v
HM G d d
�
�
1
( )
( , )
u v
d d
e uv E G
M G x x

�

�
3

Wei Gao, Mohammad Reza Farahani and Jia-Bao Liu
As a famous degree-based index, the harmonic index of molecular graph is denoted as:
Favaron et al. [21] manifested the relation between the eigenvalues and harmonic index
of molecular graphs. Zhong [22] reported the minimum and maximum values of the harmonic
index for connected molecular graphs and trees, and the corresponding extremal molecular
graphs are also described. Wu et al. [23] derived the minimum value of the harmonic index
with the minimum degree at least two. Liu [24] yielded the relationship between the diameter
and the harmonic index of molecular graphs.
Very recently, Yan et al. [25] introduced the general harmonic index for extending
harmonic index in more chemical engineering applications which can be stated as
In this paper, we focus on the structures special chemical molecular graphs and compute
these above mention degree-based topological indices.
2. MAIN RESULTS
The aim of this section is to compute the general Randić index, Randić connectivity
index, Sum-connectivity Index, general sum connectivity index, the second Zagreb index
M2(G), the first Zagreb index M1(G) and the first Hyper-Zagreb index HM1(G), the first and
the second Zagreb polynomials, general harmonic index, harmonic index for a special
chemical molecular graph “Cas(C)-CaR(C)[m,n,p] Nanotubes Junction.” The Cas(C)-
CaR(C)[m,n,p] Nanotubes Junction is a new nano-structure that was defined by M.V. Diudea
[26], on based the new graph operations on the cycle graph Cn, namely : Leapfrog Le and
Capra Ca.
2
( )
( , )
u v
d d
e uv E G
GM x x
�

�
 
( )
2
e uv E G u v
Hd d
G
�


�
 
( )
2
k
uv E G u v
k
Gd
Hd
�
� �
� �

� �

�
4

Degree-Based Indices Computation for Special Chemical Molecular and Nanotubes
Figure 1. An example of “Leapfrog Le(C6)” graph operation.
Figure 2. An example of “Capra Ca(C4)” graph operation.
Some examples of graph operations (Leapfrog Le and Capra Ca) are shown in Figures 1
and 2 and readers can see the references [27-29].
Now, consider Cas(C)-CaR(C)[m,n,p] Nanotubes Junction

m,n,p

ℕ, such that the
3-Dimensional lattice of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction are shown in Figure 3.
In this paper we name the first member Cas(C)[1,1,1] or Cas(C) as the based unit (see
Figures 3 & 4), since all member of Cas(C)[m,n,p] Nanotubes are combine this unit.
From Figure 3, one can see that 6 × 4 = 24 vertices/atoms of Cas(C) unit have degree 2
(red colored vertices in Figure 3), and there are 2 × 4 = 8 vertices/atoms with degree 3 in any
split of Cas(C) (yellow colored vertices in Figure 3) and Cas(C) unit has 6 splits. Finally
there are 8 common vertices between 3 joist splits of Cas(C) (obviously with degree 3 and
colored by white). These imply that Cas(C) unit has 24+6×8+8 = 80 (|
V(Cas(C))|)vertices/atoms and the number of edges/bonds of Cas(C) unit is equal to
|E(Cas(C))| = = ½[2×24+3×56] = 216.
Thus following M. V. Diudea [30] we denote the number of Cas(C) units in the first rows
and column in this Nanotubes by integer number m, n and p. Therefore, in general case of this
nano-structure Cas(C)-CaR(C)[m,n,p], there are m×n×p Cas(C) units and there exist |
V(Cas(C)-CaR(C)[m,n,p] )| = 80×m×n×p = 80mnp number of vertices/atoms (

m,n,p

ℕ).
2 3
2 3
2
V V� �
5

Wei Gao, Mohammad Reza Farahani and Jia-Bao Liu
Figure 3. The based unit Cas(C)-CaR(C)[1,1,1] of the Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction m,nℕ.
Figure 4. A-Dimensional lattice of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction

m,n,p

ℕ.
Also, from the structure of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction

m,n,p

ℕ, in
Figure 4, one can see that the number of edges/bonds of Cas(C)-CaR(C)[m,n,p] is equal to
|E(Cas(C)-CaR(C)[m,n,p] )| = 216×m×n×p+4(m-1)(n-1)(p-1)
= 220 mnp-4mn-4mp-4np+4m+4n+4p-4.
Now, for this we perform some necessary calculations for computing the general Randić
index, Randić connectivity index, Sum-connectivity Index, general sum connectivity index,
the second Zagreb index M2(G), the first Zagreb index M1(G) and the first Hyper-Zagreb
index HM1(G), the first and the second Zagreb polynomials, general Harmonic and Harmonic
indices defined in the previous section. Let us define the partitions for the vertex set and edge
set of the Nanotubes for δ≤k≤Δ, 2δ≤i≤2Δ, and δ2≤j≤Δ2 [31, 32], then we have
Vk = {v

V(G)| dv = k},
Ei = {e = uv

E(G)|du+dv = i},
Ej* = {uv

E(G)|du×dv = j}.
where, the vertex partitions and the edge partitions and are collectively
exhaustive, that is
In the case G = Cas(C) unit, one can see that ∀v∊ V(Cas(C) dv = 2 or 3. Thus, we have
the vertex partitions with their cardinalities as follows.
k
V
i
E
*
j
E
).(=),(=),(=
*
)(
2
)(
2
=
)(2
)(2=
)(
)(=
GEEGEEGVV
j
G
Gj
i
G
Gi
k
G
Gk




6

Degree-Based Indices Computation for Special Chemical Molecular and Nanotubes
V3 = {v

V(Cas(C))| dv = 3}
V2 = {v

V(Cas(C))| dv = 2}
Vertex partition V3V2
Cardinality 56 24
And the edge partitions of Cas(C) unit with their cardinalities are as follows.
E5 = E6* = {uvE(Cas(C))|du = 2 & dv = 3}
E6 = E9* = {uvE(Cas(C))|du = dv = 3}.
Edge partition E5 = E6*E6 = E9*
Cardinality 2×| V2| = 48 168
In the general case G = Cas(C)-CaR(C)[m,n,p] Nanotubes Junction, one can see that
∀v∊ V(Cas(C)-CaR(C)[m,n,p] ) dv = 2 or 3, too and we have the vertex and edge partitions
with their cardinalities as follows (

m,n,p

ℕ).
V3 = {vV(Cas(C)-CaR(C)[m,n,p] )| dv = 3}
V2 = {vV(Cas(C)-CaR(C)[m,n,p] )| dv = 2}
Vertex partition V2V3
Cardinality 4(2mp+2np+2mn) 8(10mnp-mp-np-mn)
E5 = E6* = {uvE(Cas(C)-CaR(C)[m,n,p] )|du = 2 & dv = 3}
E6 = E9* = {uvE(Cas(C)-CaR(C)[m,n,p] )|du = dv = 3}.
Edge partition E5 = E6*E6 = E9*
Cardinality 8(2mp+2np+2mn) |E(Cas(C)-CaR(C)[m,n,p])|-16(mp+np+mn)
= 220mnp-20mn-20mp-20np+4m+4n+4p-4
Here, according to the definitions of above mention degree-based indices in Section 1, we
deduce:
The general Randić index in the general case G = Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction is equal to
Rk(Cas(C)-CaR(C)[m,n,p] ) =
= |E9*|(3×3)k+|E6*|(2×3)k
= 32k×4(55mnp-5mn-5mp-5np+m+n+p-1)+16(6k)(mp+np+mn)
The Randić connectivity index of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction is equal to
 
 
 
( , , )
k
u v
e uv E Cas C m n p
d d
�
�
   
1 1 2 2
* *
1 1 9 2 2 6
u v u v
k k
u v E u v E
d d d d
� �
 
� �
7

Wei Gao, Mohammad Reza Farahani and Jia-Bao Liu

(Cas(C)-CaR(C)[m,n,p] ) =
[ (55mnp-5mn-5mp-5np+m+n+p-1)+ (mp+np+mn)]
[ 55mnp-( )(mp+np+mn)+m+n+p-1].
The second Zagreb and Hyper-Zagreb indices of Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction are equal to
M2(Cas(C)-CaR(C)[m,n,p] ) =
= |E9*|(3×3)+|E6*|(2×3)
= 36(55mnp-5mn-5mp-5np+m+n+p-1)+96(mp+np+mn)
= 1980mnp-84(mp+np+mn)+36(m+n+p-1).
HM2(Cas(C)-CaR(C)[m,n,p] ) = (dvdv)2
= |E9*|(3×3)2+|E6*|(2×3)2
= 81(55mnp-5mn-5mp-5np+m+n+p-1)+36×16(mp+np+mn)
= 4455mnp+171(mp+np+mn)+81(m+n+p-1).
Here the general sum connectivity index in the general case G = Cas(C)-CaR(C)[m,n,p]
Nanotubes Junction is equal to
Xk(G) =
= |E6|(3+3)k+|E5|(2+3)k
= 6k×4(55mnp-5mn-5mp-5np+m+n+p-1)+(5k)×16 (mp+np+mn).
And obviously the sum connectivity index of Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction is
X(Cas(C)-CaR(C)[m,n,p] ) =
 
 
( , , )
1
e uv E Cas C m n p u v
d d
�
�
1 1 2 2
* *
1 1 9 2 2 6
1 1
u v u v
u v E u v E
d d d d
� �
 
� �
* *
9 6
6
3 6
E E
 
3
4
62
3
4
5 62
 
 
 
( , , )
u v
uv E Cas C m n p
d d
�
�
�
   
* *
9 6
u v u v
uv E uv E
d d d d
� �
 
� �
 
 
 
, ,Cas Ce mE n p�
�
 
( )
u v
uv
k
E G
d d
�

�
   
1 1 2 2
5 6
1 1 2 2
k
v u v
k
u
u v E u v E
d d d d
� �
  
� �
 
 
( , , )
1
uv E Cas C m n p u v
d d
�

�
8

Degree-Based Indices Computation for Special Chemical Molecular and Nanotubes
(55mnp-5mn-5mp-5np+m+n+p-1)+ (mp+np+mn).
The first Zagreb and Hyper-Zagreb indices of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction
are equal to
M1(Cas(C)-CaR(C)[m,n,p] ) =
= 5|E5|+6|E6|
= 80(mp+np+mn)+24(55mnp-5mn-5mp-5np+m+n+p-1)
= 1320mnp-40(mp+np+mn)+24(m+n+p-1)
and
HM1(Cas(C)-CaR(C)[m,n,p] ) = (dv+dv)2
= 25|E5|+36|E6|
= 400(mp+np+mn)+144(55mnp-5mn-5mp-5np+m+n+p-1)
= 7920mnp-320(mp+np+mn)+144(m+n+p-1)
= 16(495mnp-20(mp+np+mn)+9(m+n+p-1).
Here by according to the definition of the general harmonic index of a molecular graph
G, we see that
Hk(G) = = 2kX(-k)(G)
Thus the general harmonic index in the general case Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction is equal to
Hk(Cas(C)-CaR(C)[m,n,p] ) = 2k6(-k)×4(55mnp-5mn-5mp-5np+m+n+p-1)
+2k5(-k)×16(mp+np+mn)
Also, this implies that the harmonic index of G = Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction is equal to
5 6
1 1
u v u v
uv E uv E
d d d d
� �
 
 
� �
5 6
5
| |
6
| |E E
 
2
3
6
16
5
5

 
 
 
( , , )
u v
uv E Cas C m n p
d d
�

�
   
5
6
u v u v
uv E uv E
d d d d
��
  
� �
 
 
 
, ,Cas Ce mE n p�
�
   
5
2 2
6
v v v v
uv E uv E
d d d d
��
   
� �
( )
2
k
uv E G u v
d d
�
� �
� �

� �
�
9

Wei Gao, Mohammad Reza Farahani and Jia-Bao Liu
H(G) =
= 6.4(mp+np+mn)+ (55mnp-5mn-5mp-5np+m+n+p-1)
≈73.333mnp-0.2666(mp+np+mn)+1.333(m+n+p-1).
CONCLUSION
In this paper, a special chemical molecular graph “Cas(C)-CaR(C)[m,n,p] Nanotubes
Junction” (

m,n,p

ℕ, a 3-Dimensional lattice of Cas(C)-CaR(C)[m,n,p] Nanotubes Junction
are shown in Figure 3) has been investigated here, and formulas for computing its topological
indices “general harmonic, harmonic, general Randić, Randić connectivity, Sum-connectivity,
general sum connectivity, first and second Zagreb, first and second Hyper Zagreb, first and
second Zagreb polynomials, indices” and the corresponding Zagreb polynomials have been
derived.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this
paper.
ACKNOWLEDGMENTS
The authors are thankful to Prof. M. V. Diudea and Prof. Csaba Nagy from Faculty of
Chemistry and Chemical Engineering Babes-Bolyai University (Romania) for helpful
comments and suggestions.
Project Supported by the Natural Science Foundation for the Higher Education
Institutions of Anhui Province of China (Grant Nos. KJ2015A178, KJ2015A256,
KJ2015A331).
REFERENCES
[1] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH,
Weinheim, 2000.
[2] N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992.
[3] L. B. Kier, L. H. Hall, Molecular Connectivity in Chemistry and Drug Research.
Academic Press, New York, 1976.

 
)(
2
GEuve vu dd
5
6
2 2
u v u v
uv E uv E
d d d d
��
 
� � � �
� � � �
 
� � � �
� �
5 6
2 2
5
|
6
| | |E E
 
4
3

10

Degree-Based Indices Computation for Special Chemical Molecular and Nanotubes
[4] L. B. Kier, L. H. Hall, Molecular Connectivity in Structure-Activity Analysis. Research
Studies Press/Wiley, Letchworth/New York, 1986.
[5] H. Wiener, J. Amer. Chem. Soc. 1947, 69, 17-20.
[6] B. Zhou and I. Gutman, Chemical Physics Letters. 2004, 394, 93-95.
[7] A. A. Dobrynin, R. Entringer and I. Gutman, Acta Appl. Math. 2001, 66, 211.
[8] D. E. Needham, I. C. Wei and P. G. Seybold, J. Amer. Chem. Soc. 1988, 110, 4186.
[9] G. Rucker and C. Rucker, J. Chem. Inf. Comput. Sci. 1999, 39, 788.
[10] M. Randić. J. Amer. Chem. Soc, 1975, 97, 6609-6615.
[11] B. Bollobas, P. Erdos. Ars Combinatoria, 1998, 50 225-233.
[12] B. Li, W. Liu. Proceedings of the Indian Academy of Sciencesmathematical Sciences,
2013, 123(2) 167-175.
[13] B. L. Liu, I. Gutman. MATCH Communications in Mathematical and in Computer
Chemistry, 2007, 57 617-632.
[14] B. Zhou and N. Trinajstić, J. Math. Chem. 2009, 46, 1252-1270.
[15] B. Zhou, N. Trinajstić. Journal of Mathematical Chemistry, 2010, 47 210-218.
[16] Gutman and K. C. Das, MATCH Commun. Math. Comput. Chem, 2004, 50, 83-92.
[17] Gutman and N. Trinajstć, Chemical Physics Letters, 1972, 17, 535-538.
[18] Gutman, B. Ruscic, N. Trinajstić and C. F. Wilcox, J. Chem. Phys, 1975, 62, 3399-
3405.
[19] Ilicć and D. Stevanović, MATCH Commun. Math. Comput. Chem, 62, (2009), 681-687.
[20] G. H. Shirdel, H. RezaPour, A. M. Sayadi. Iranian Journal of Mathematical Chemistry,
2013, 4(2) 213-220.
[21] O. Favaron, M. Maho, J. F. Sacle. Discrete Mathematics, 1993, 111(1-3) 197-220.
[22] L. Zhong. Applied Mathematics Letters, 2012, 25(3) 561-566.
[23] R. Wu, Z. Tang, H. Deng. Filomat, 2013, 27(1), 51-55.
[24] J. X. Liu. Journal of Applied Mathematics and Physics, 2013, 1, 5-6.
[25] L. Yan, W. Gao, J. S. Li. Journal of Computational and Theoretical Nanoscience, 2015,
In press.
[26] M. V. Diudea. Nanostructuri Periodice. 2004, Online manuscript.
[27] M. V. Diudea, (Ed.), QSPR/QSAR Studies by Molecular Descriptors, NOVA, New
York, 2001.
[28] M. V. Diudea, I. Gutman and L. Jäntschi, Molecular Topology, NOVA, New York,
2002.
[29] M. V. Diudea, M. S. Florescu and P. V. Khadikar, Molecular Topology and Its
Applications, EFICON, Bucharest, 2006.
[30] M. V. Diudea, Fuller. Nanotub. Carbon Nanostruct, 2002, 10, 273.
[31] M. R. Farahani. Acta Chim. Slov. 2012, 59, 779-783.
[32] W. Gao and M. R. Farahani. Applied Mathematics and Nonlinear Sciences. 2015, 1(1),
94–117.
11