Multiplicative topological descriptors of Silicon carbide

Open access

Abstract

Topological indices helps us to collect information about algebraic graphs and gives us mathematical approach to understand the properties of chemical structures. In this paper, we aim to compute multiplicative degree-based topological indices of Silicon-Carbon Si 2 C 3 −III[p,q] and SiC 3 −III[p,q] .

Abstract

Topological indices helps us to collect information about algebraic graphs and gives us mathematical approach to understand the properties of chemical structures. In this paper, we aim to compute multiplicative degree-based topological indices of Silicon-Carbon Si2C3−III[p,q] and SiC3−III[p,q] .

1 Introduction

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pair-wise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics and found many applications in our life [1, 2, 3, 4].

In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound [5, 6, 7, 8]. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development

of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure [9, 10, 11, 12].

In mathematical chemistry, precisely speaking, in chemical-graph-theory (CGT), a molecular graph and graph network is a simple and connected graph, in which atoms represents vertices and chemical bonds represents edges. We reserve G for simple connected graph, E for edge set and V for vertex set throughout the thesis. The degree of a vertex u of graph G is the number of vertices that are attached with u and is denoted by dv. With the help of TIs, many properties of molecular structure can be obtained without going to lab [13]. The reality is, many research paper has been written on computation of degree-based indices and polynomials of different molecular structure and networks but only few work has been done so far on distance based indices and polynomials. In this paper, we aim to compute multiplicative degree-based TIs. Some indices related to Wiener’s work are the first and second multiplicative Zagreb indices [14], respectively

II1(G)=uV(G)(du)2,II2(G)=uvE(G)dudv.

and the Narumi-Katayama index [52]

NK(G)=uV(G)du.

Like the Wiener index, these types of indices are the focus of considerable research in computational chemistry [16, 17, 18]. For example, in the year 2011, Gutman in [16] characterized the multiplicative Zagreb indices for trees and determined the unique trees that obtained maximum and minimum values for M1(G) and M2(G), respectively. Wang et al in [19] extended the results of Gutman to the following index for k-trees,

W1s(G)=uV(G)(du)s.

Notice that s = 1,2 is the Narumi-Katayama and Zagreb index, respectively. Based on the successful consideration of multiplicative Zagreb indices, Eliasi et al [20] continued to define a new multiplicative version of the first Zagreb index as

II1*(G)=uvE(G)(du+dv).

Furthering the concept of indexing with the edge set, the first author introduced the first and second hyper-Zagreb indices of a graph [21]. They are defined as

HII1(G)=uv(G)(du+dv)2,HII2(G)=uvE(G)(dudv)2.

In [22] Kulli et al defined the first and second generalized Zagreb indices

MZ1a(G)=uv(G)(du+dv)α,MZ2a(G)=uvE(G)(dudv)α.

Multiplicative sum connectivity and multiplicative product connectivity indices [23] are define as:

SCII(G)=uvE(G)1du+dv,

PCII(G)=uvE(G)1dudv.

Multiplicative atomic bond connectivity index and multiplicative Geometric arithmetic index are defined as

ABCII(G)=uvE(G)du+dv2dudv,GAII(G)=uvE(G)2dudvdu+dv,GAaII(G)=uvE(G)(2dudvdu+dv)α.

2 Silicon Carbide

In 1891, an American scientist discover Silicon Carbide. But now a days, we can produce silicon carbide artificially by silica and carbon. Till 1929, silicon carbide was known as the hardest material on earth.Its Mohs hardness rating is 9, which makes this similar to diamond. Here, we will find out reverse zagreb, hyper reverse zagreb and its polynomials for silicon carbide Si2C3 −III[p,q] and SiC3 −III[p,q]. We consider 2D SiC compounds with two different types of SiC structure based on low-energy metastable structures for each SiC. The types are Si2C3−III[p,q] and SiC3−III[p,q] that denotes the lowest-energy and the second lowest energy structure respectively. The unit cell of Si2C3−III[p,q] is given in Figure 1. The 2D lattice graphs of Si2C3−III[5,1], Si2C3−III[5,2] and Si2C3−III[5,4] are shown in Figures2,3 and 4 respectively.

Fig. 1

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Fig. 1

Unit Cell

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 2

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Fig. 2

Si2C3−III[5,1]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 3

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Fig. 3

Si2C3−III[5,2]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 4

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Fig. 4

Si2C3−III[5,4]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 5

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Fig. 5

Unit Cell

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 6

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Fig. 6

SiC3−III[5,1]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Fig. 7

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Fig. 7

SiC3−III[5,2]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

3 Methodology

To compute our main results we count the number of edges of Si2C3−III[p,q] and SiC3−III[p,q] by using Figures 1-4 and Figures 5-8 respectively. After that, we divide these edge sets into classes based on the degree of vertices. The Edge partition of Si2C3−III[p,q] is given in Table 1 and the edge partition of SiC3−III(G) is given in Table 2. By using these edge partitions, we compute our main results.

Fig. 8

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Fig. 8

SiC3−III[5,4]

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00018

Table 1

Edge partition of Si2C3−III[p,q]

(du,dv)Frequency
(1,3)2
(2,2)2p+2
(2,3)8p+8q-12
(3,3)15pq-10p-13q+8
Table 2

Edge partition of SiC3−III[p,q]

(du, dv)Frequency
(1,2)2
(1,3)1
(2,2)3p+2q-3
(2,3)6p+4q-8
(3,3)12pq-12p-8q+8

4 Main Results

In this article, we will compute some degree-based multiplicative topological indices for Si2C3 −III[p,q] and SiC3−III[p,q].

4.1 Multiplicative Topological Indices for Silicon Carbides Si2C3−III[p,q]

Theorem 4.2

Let Si2C3−III[p,q] be the Silicon Carbide. Then

1.MZ1α(Si2C3III[p,q])=(2)α(15pq10p5q+16)×(3)α(15pq10p13q+8)×(5)α(8p+8q12).2.MZ2α(Si2C3III[p,q])=(2)4α(2p+3q2)×(3)6α(5pq2p3q+1).3.GAαII(Si2C3III[p,q])=(2)2α(6p+6q5)×(3)α(4p+4q5)×(5)α(128q8p).

Proof Using edge partition given in Table 1, we have

MZ1α(Si2C3III[p,q])=uvεE(Si2C3III[p,q])(du+dv)α=(1+3)2α×(2+2)α(2q+2)×(2+3)α(8p+8q12)×(3+3)α(15pq10p13q+8)=(2)α(15pq10p5q+16)×(3)α(15pq10p13q+8)×(5)α(8p+8q12).

MZ2α(Si2C3III[p,q])=uvεE(Si2C3III[p,q])(du+dv)α=(1×3)2α×(2×2)α(2q+2)×(2×3)α(8p+8q12)×(3×3)α(15pq10p13q+8)=(2)4α(2p+3q12)×(3)6α(5pq2p3q+1).

GAαII(Si2C3III[p,q])=uvεE(Si2C3III[p,q])(2du×dvdu+dv)α=(21×31+3)2α×(22×22+2)α(2q+2)(22×32+3)α(8p+8q12)×(23×33+2)α(15pq10p13q+8)=(2)2α(6p+6q5)×(3)α(4p+4q5)×(5)α(128q8p).

Theorem 4.3

Let Si2C3−III[p,q] be the Silicon Carbide. Then

1.MZ1(Si2C3III[p,q])=II0*=(2)(15pq10p5q+16)×(3)(15pq10p13q+8)×(5)(8p+8q12).

  1. MZ2(Si2C3−III[p,q]) = (2)4(2p+3q−2)×(3)6(5pq−2p−3q+1).

  2. GAII(Si2C3−III[p,q]) = (2)2(6p+6q−5)×(3)(4p+4q−5)×(5)(128q−8p).

Proof Taking α = 1, in Theorem 4.2, we get our desire results.

Theorem 4.4

Let S2iC3−III[p,q]II[p,q] be the Silicon Carbide. Then

  1. HII1(Si2C3−III[p,q]) = (2)2(15pq−10p−5q+16)×(3)2(15pq−10p−13q+8)×(5)2(8p+8q−12)

  2. HII2(Si2C3−III[p,q]) = (2)8(2p+3q−2)×(3)12(5pq−2p−3q+1)

Proof Taking α = 2 in Theorem 4.2, we get our desire results.

Theorem 4.5. Let Si2C3−III[p,q] be the Silicon Carbide. Then

1.SCII(Si2C3III[p,q])=(12)(15pq10p5q+16)×(13)(15pq10p13q+8)×(15)(8p+8q12).2.PCII(Si2C3III[p,q])=(12)4(2p+3q2)×(13)6(5pq2p3q+1).

Proof Taking α=12in Theorem 4.2, we get our desire results.

Theorem 4.6

Let Si2C3−III[p,q] be the Silicon Carbide. Then

ABCII(Si2C3III[p,q])=[(12)12]4p+5q5×[(23)]15pq10p13q+9.

Proof

GAαII(Si2C3III[p,q])=uvεE(Si2C3III[p,q])du+dv2du×dv                                                      =(1+321×3)2×(2+222×2)2q+2                                                       ×(2+322×3)(8p+8q12)×(3+323×3)(15pq10p13q+8)                                                      =[(12)12]4p+5q5×[(23)]15pq10p13q+9.

4.7 Multiplicative Topological Indices for Silicon Carbides SiC3−III[p,q]

Theorem 4.8

Let SiC3−III[p,q] be the Silicon Carbide. Then

1.MZ1α(Si2C3III[p,q])=(2)α(12pq+6p+4q+4)×(3)α(12pq12p8q+10)×(5)α(6p+4q8).2.MZ2α(Si2C3III[p,q])=(2)4α(3p+2q3)×(3)3α(8pq6p4q+3).3.GAαII(Si2C3III[p,q])=(2)2α(9p+6q10)×(3)α(3p+2q112)×(5)α(86p4q).

Proof Using the edge partition given in Table 2, we have

MZ1α(Si2C3III[p,q])=uvεE(Si2C3III[p,q])(du+dv)α=(1+2)2α×(1+3)α×(2+2)α(3p+2q3)×(2+3)α(6p+4q8)×(3+3)α(12pq12p8q+8)=(2)α(12pq+6p+4q+4)×(3)α(12pq12p8q+10)×(5)α(6p+4q8).

2.MZ2α(SiC3III[p,q])=uvεE(SiC3III[p,q])(du×dv)α=(1×2)2α×(1×3)α×(2×2)α(3p+2q3)×(2×3)α(6p+4q8)×(3×3)α(12pq12p8q+8)=(2)4α(3p+2q3)×(3)3α(8pq6p4q+3).

3.GAαII(SiC3-III[p,q])=uvεE(Si2C3-III[p,q])(2du×dvdu+dv)α=(21×21+2)2α×(21×31+3)α×(22×22+2)α(3p+2q-3)(22×32+3)α(6p+4q-8)×(23×33+2)α(12pq-12p-8q-8)=(2)α(9p+6q-10)×(3)α(3p+2q-112)×(5)α(8-6p-4q).

Theorem 4.9

Let SiC3−III[p,q] be the Silicon Carbide. Then

1.MZ1(SiC3III[p,q])=II1*=(2)(12pq+6p+4q+4)×(3)(12pq12p8q+10)×(5)(6p+4q8).2.MZ2(SiC3III[p,q])=(2)4(3p+2q3)×(3)3(8pq6p4q+3).3.GAII(SiC3III[p,q])=(2)(9p+6q10)×(3)(3p+2q112)×(5)(86p45q).

Proof Taking α = 1, in Theorem 4.8, we get our desire results.

Theorem 4.10. Let SiC3−III[p,q] be the Silicon Carbide. Then

  1. 1. HII1(SiC3−III[p,q]) = (2)2(12pq+6p+4q+4)×(3)2(12pq−12p−8q+10)×(5)2(6p+4q−8).

  2. 2. HII2(SiC3−III[p,q]) = (2)8(3p+2q−3)×(3)6(8pq−6p−4q+3).

Proof Taking α = 2, in Theoem 4.8, we get our desire results.

Theorem 4.11

Let SiC3−III[p,q] be the Silicon Carbide. Then

1.SCII(SiC3III[p,q])=(12)(12pq+6p+4q+4)×(13)α(12pq12p8q+10)×(15)α(6p+4q8).2.PCII(SiC3III[p,q])=(12)4(3p+2q3)×(13)3(8pq6p4q+3).

Proof Taking α=12,in Theorem 4.8, we get our desire results.

Theorem 4.12

Let SiC3−III[p,q] be the Silicon Carbide. Then

ABCII(SiC3III[p,q])=[16][(12)(12)]9p+6q11×[(23)]12pq12p8q+8.

Proof

GAαII(SiC3III[p,q])=uvεE(Si2C3III[p,q])du+dv2du×dv=(1+221×2)2×(1+321×3)2×(2+222×2)3p+2q3×(2+322×3)(6p+4q8)×(3+323×3)(12pq12p8q+8)=[16]×[(12)12]9p+6q11×[(23)]15pq12p8q+8.

Conclusion

Topological indices has many applications in chemistry, physics and other applied sciences. In this paper we have computed multiplicative Zagreb indices, multiplicative geometric arithmetic index, multiplicative atomic bond connectivity index, etc of two Silicon Carbide structures.

Communicated by Juan Luis García Guirao

References

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    M. A. Umar, M. A. Javed, M. Hussain, B. R. Ali. Super ( a , d ) - C 4 -antimagicness of book graphs. Open J. Math. Sci., Vol. 2(2018),No.1, pp. 115-121.

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    Girish V R, P. Usha. Secure Domination in Lict Graphs. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 134 - 145.

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    H. M. Nagesh, M. C. Mahesh Kumar. Block digraph of a directed graph. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 202 - 208.

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    M. C. Mahesh Kumar and H. M. Nagesh. Directed Pathos Total Digraph of an Arborescence. Engineering and Applied Science Letters, 1 (1), pp. 29 - 42.

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    G. Liu, Z. Jia, W. Gao. Ontology similarity computing based on stochastic primal dual coordinate technique. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 221 - 227.

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    M.S. Sardar, S. Zafar, M.R. Farahani. The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series C a k ( C 6 ). Open J. Math. Sci., Vol. 1(2017), No. 1, pp. 44 - 51

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    M.S. Sardar, X.-F. Pan, W. Gao, M.R. Farahani. Computing Sanskruti Index of Titania Nanotubes. Open J. Math. Sci., Vol. 1(2017), No. 1, pp. 126 - 131

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    H. Siddiqui, M.R. Farahani. Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks. Open J. Math. Anal., Vol. 1(2017), Issue 1, pp. 45-60.

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    W. Gao, B. Muzaffar, W. Nazeer. K-Banhatti and K-hyper Banhatti Indices of Dominating David Derived Network. Open J. Math. Anal., Vol. 1(2017), Issue 1, pp. 13-24.

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    S. Noreen, A. Mahmood. Zagreb Polynomials and Redefined Zagreb Indices for the line graph of Carbon Nanocones. Open J. Math. Anal., Vol. 2(2018), Issue 1, pp. 67-76.

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    W. Gao, M. Asif, W. Nazeer. The Study of Honey Comb Derived Network via Topological Indices. Open J. Math. Anal., Vol. 2(2018), Issue 2, pp. 10-26.

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    Rehman, H. M., Sardar, R., & Raza, A. (2017). Computing topological indices of hex board and its line graph. Open J. Math. Sci., 1, 62-71.

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    Kwun, Y. C., Munir, M., Nazeer,W., Rafique, S., & Kang, S. M. (2018). Computational Analysis of topological indices of two Boron Nanotubes. Scientific reports 8 1, 14843.

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    Li, X. & Gutman, I. (2006). Mathematical Chemistry Monographs, No 1 Kragujevac.

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    Narumi, H., & Katayama, M. (1984). Simple topological index: A newly devised index characterizing the topological nature of structural isomers ofsaturated hydrocarbons. Memoirs of the Faculty of Engineering, Hokkaido University 16(3), 209-214.

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    Gutman, I. (2011). Multiplicative Zagreb indices of trees. Bull. Soc. Math. Banja Luka, 18 17-23.

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    Todeschini, R., & Consonni, V. (2010). New local vertex invariants and molecular descriptors based on functions of the vertex degrees. MATCH Commun. Math. Comput. Chem, 64(2), 359-372.

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    Wang, S., & Wei, B. (2015). Multiplicative Zagreb indices of k-trees. Discrete Applied Mathematics, 180 168-175.

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    Kulli, V. R. (2016). Multiplicative connectivity indices of TUC4C8[m,n] and TUC4[m,n] nanotubes. Journal of Computer and Mathematical Sciences, 7(11), 599-605.

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    Shigehalli, V. S., & Kanabur, R. (2016). Computation of New Degree-Based Topological Indices of Graphene. Journal of Mathematics, 2016 Article ID 4341919, 6 pages.

[1]

M. A. Umar, M. A. Javed, M. Hussain, B. R. Ali. Super ( a , d ) - C 4 -antimagicness of book graphs. Open J. Math. Sci., Vol. 2(2018),No.1, pp. 115-121.

[2]

Girish V R, P. Usha. Secure Domination in Lict Graphs. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 134 - 145.

[3]

H. M. Nagesh, M. C. Mahesh Kumar. Block digraph of a directed graph. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 202 - 208.

[4]

M. C. Mahesh Kumar and H. M. Nagesh. Directed Pathos Total Digraph of an Arborescence. Engineering and Applied Science Letters, 1 (1), pp. 29 - 42.

[5]

G. Liu, Z. Jia, W. Gao. Ontology similarity computing based on stochastic primal dual coordinate technique. Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 221 - 227.

[6]

M.S. Sardar, S. Zafar, M.R. Farahani. The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series C a k ( C 6 ). Open J. Math. Sci., Vol. 1(2017), No. 1, pp. 44 - 51

[7]

M.S. Sardar, X.-F. Pan, W. Gao, M.R. Farahani. Computing Sanskruti Index of Titania Nanotubes. Open J. Math. Sci., Vol. 1(2017), No. 1, pp. 126 - 131

[8]

H. Siddiqui, M.R. Farahani. Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks. Open J. Math. Anal., Vol. 1(2017), Issue 1, pp. 45-60.

[9]

W. Gao, B. Muzaffar, W. Nazeer. K-Banhatti and K-hyper Banhatti Indices of Dominating David Derived Network. Open J. Math. Anal., Vol. 1(2017), Issue 1, pp. 13-24.

[10]

S. Noreen, A. Mahmood. Zagreb Polynomials and Redefined Zagreb Indices for the line graph of Carbon Nanocones. Open J. Math. Anal., Vol. 2(2018), Issue 1, pp. 67-76.

[11]

W. Gao, M. Asif, W. Nazeer. The Study of Honey Comb Derived Network via Topological Indices. Open J. Math. Anal., Vol. 2(2018), Issue 2, pp. 10-26.

[12]

Rehman, H. M., Sardar, R., & Raza, A. (2017). Computing topological indices of hex board and its line graph. Open J. Math. Sci., 1, 62-71.

[13]

Kwun, Y. C., Munir, M., Nazeer,W., Rafique, S., & Kang, S. M. (2018). Computational Analysis of topological indices of two Boron Nanotubes. Scientific reports 8 1, 14843.

[14]

Li, X. & Gutman, I. (2006). Mathematical Chemistry Monographs, No 1 Kragujevac.

[15]

Narumi, H., & Katayama, M. (1984). Simple topological index: A newly devised index characterizing the topological nature of structural isomers ofsaturated hydrocarbons. Memoirs of the Faculty of Engineering, Hokkaido University 16(3), 209-214.

[16]

Gutman, I. (2011). Multiplicative Zagreb indices of trees. Bull. Soc. Math. Banja Luka, 18 17-23.

[17]

Todeschini, R., Ballabio, D., & Consonni, V. (2010). Novel molecular descriptors based on functions of new vertex degrees. Mathematical Chemistry Monographs, 2010 73-100.

[18]

Todeschini, R., & Consonni, V. (2010). New local vertex invariants and molecular descriptors based on functions of the vertex degrees. MATCH Commun. Math. Comput. Chem, 64(2), 359-372.

[19]

Wang, S., & Wei, B. (2015). Multiplicative Zagreb indices of k-trees. Discrete Applied Mathematics, 180 168-175.

[20]

Eliasi, M., Iranmanesh, A., & Gutman, I. (2012). Multiplicative versions of first Zagreb index. Match-Communications in Mathematical and Computer Chemistry, 68(1), 217.

[21]

Kulli, V. R. (2016). Multiplicative hyper-Zagreb indices and coindices of graphs: computing these indices of some nanostructures. International Research Journal of Pure Algebra, 6(7), 342-347.

[22]

Kulli, V. R., Stone, B., Wang, S., & Wei, B. (2017). Generalised multiplicative indices of polycyclic aromatic hydrocarbons and benzenoid systems. Zeitschrift für Naturforschung A, 72(6), 573-576.

[23]

Kulli, V. R. (2016). Multiplicative connectivity indices of TUC4C8[m,n] and TUC4[m,n] nanotubes. Journal of Computer and Mathematical Sciences, 7(11), 599-605.

[24]

Shigehalli, V. S., & Kanabur, R. (2016). Computation of New Degree-Based Topological Indices of Graphene. Journal of Mathematics, 2016 Article ID 4341919, 6 pages.

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