Revan and hyper-Revan indices of Octahedral and icosahedral networks

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Abstract

Let G be a connected graph with vertex set V(G) and edge set E(G). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of G are defined as R1(G) = uvE[rG(u) + rG(v)] and R2(G) = uvE[rG(u)rG(v)], where uv means that the vertex u and edge v are adjacent in G. The first and second hyper-Revan indices of G are defined as HR1(G) = uvE[rG(u) + rG(v)]2 and HR2(G) = uvE[rG(u)rG(v)]2. In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.

Abstract

Let G be a connected graph with vertex set V(G) and edge set E(G). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of G are defined as R1(G) = uvE[rG(u) + rG(v)] and R2(G) = uvE[rG(u)rG(v)], where uv means that the vertex u and edge v are adjacent in G. The first and second hyper-Revan indices of G are defined as HR1(G) = uvE[rG(u) + rG(v)]2 and HR2(G) = uvE[rG(u)rG(v)]2. In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.

1 Introduction

Topological indices of large chemical structures such as metal organic frameworks can be extremely useful in both characterization of structures and computing their physico-chemical properties that are otherwise difficult to compute for such large networks of importance in reticular chemistry. Synthesis of novel reticular metal organic frameworks and networks in which covalent fibers weaved into crystals are becoming increasingly important in recent years [1], [2]. A topological index is a numerical parameter mathematically derived from the graph structure. Numerous topological indices have been considered in Theoretical Chemistry and many topological indices are defined by using vertex degree concept. Among them, topological descriptors are of great importance, as they deal with topological characterizations of the molecules. In general, it acts like advanced models in the chemical and control field of applications.

The concept of topological index originated from the pioneering work of Wiener while he was attempting to find structural relationships to boiling points of paraffins. There are a large number of topological indices which are classified based on the structural properties of the graphs used for their calculations. In general, they are classified into distance-based topological indices, degree-based topological indices and counting related indices of graphs. The degree-based topological indices, which have reliable predicting power and thus they have been employed in deriving multi-linear regression models for statistical correlation of properties. The topological indices such as atom-bond connectivity and geometric-arithmetic are also used to predict the bio-activity of the chemical compounds. These classes of topological indices are of great importance and play a vital role in chemical characterization. More related contexts can refer to Zhao et al. [3], Basavanagoud et al. [4], Dobrynin et al. [5], Imran et al. [6] and Sardar et al. [7].

Very recently, Kulli defined a novel degree concept in graph theory: the Revan vertex degree and determined exact formulae for oxide and honeycomb networks. We consider only finite, simple and connected graph G. Let δ(G) denote the minimum degree and Δ(G) denote the maximum degree of graph G. The Revan vertex degree of a vertex v in G is defined as rG (v) = Δ(G) + δ(G) – dG(v). The first and second Revan indices of G are defined as:

R1(G)=uvE[rG(u)+rG(v)],

and

R2(G)=uvE[rG(u)rG(v)],

where uv means that the vertex u and edge v are adjacent in G. The first and second hyper Revan indices of G are defined as

HR1(G)=uvE[rG(u)+rG(v)]2

HR2(G)=uvE[rG(u)rG(v)]2

where rG (v) = Δ(G) + δ(G) –dG (v). We refer to [8] for details about these indices.

An octahedral sheet-like structure is a ring of octahedral structures which are linked to other rings by sharing corner vertices in a two dimensional plane. An octahedral network of dimension n is denoted by OTn, where n is the order of circumscribing. The numbers of vertices and edges in OTn with n ≥ 1 are 27n2 + 3n and 72n2 respectively. We study the Revan indices for the octahedral network as follows.

Theorem 1

Let G = OTn be the octahedral network. Then the first and the second Revan indices of OTn are

R1[OTn]=864n2+96n,R2[OTn]=2592n2+576n.

Proof

Let G = OTn be the octahedral network. From figure 1, the edge partition of octahedral network OTn based on degrees of end vertices of each edge is given in table 1. First Revan index of OTn is calculated as

R1[OTn]=uvE[rG(u)+rG(v)]=uvE4,4[rG(u)+rG(v)]+uvE4,8[rG(u)+rG(v)]+uvE8,8[rG(u)+rG(v)]=(18n2+12n)[8+8]+36n2[8+4]+(18n212n)[4+4]=864n2+96n

Fig. 1

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Fig. 1

Octahedral network OT3

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

Table 1

Edge partition of octahedral network

(du, dv)Number of edgesrG(u)rG(v)
(4,4)18n2 + 12n88
(4,8)36n284
(8,8)18n2 — 12n44

Second Revan index of OTn is calculated as

R2[OTn]=ue[rG(u)rG(v)]=uvE4,4[rG(u)rG(v)]+uvE4,8[rG(u)rG(v)]+uvE8,8[rG(u)rG(v)]=(18n2+12n)[8×8]+36n2[8×4]+(18n212n)[4×4]=2592n2+576n.

Fig. 2

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Fig. 2

The green and blue sheets show the comparison result for Revan index of first and second kind Octahedral network OTn respectively.

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

Theorem 2

Let G = OTn be the octahedral network. Then the first and the second hyper Revan indices of OTn are

HR1[OTn]=10944n2+2304n,HR2[OTn]=115200n2+46080n.

Proof

Let G = OTn be the octahedral network. Then first hyper Revan index of OTn is calculated as

HR1[OTn]=uvE[rG(u)+rG(v)]2=uvE4,4[rG(u)+rG(v)]2+uvE4,8[rG(u)+rG(v)]2+uvE8,8[rG(u)+rG(v)]2=(18n2+12n)[8+8]2+36n2[8+4]2+(18n212n)[4+4]2=10944n2+2304n.

Second hyper Revan index of OTn is calculated as

HR2[OTn]=uv[rG(u)rG(v)]2=uvE4,4[rG(u)rG(v)]2+uvE4,8[rG(u)rG(v)]2+uvE8,8[rG(u)rG(v)]2=(18n2+12n)[8×8]2+36n2[8×4]2+(18n212n)[4×4]2=115200n2+46080n.

Fig. 3

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Fig. 3

The green and blue sheets show the comparison result for hyper Revan index of first and second kind Octahedral network OTn respectively.

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

Now we introduce another new network based on icosahedron. Icosahedral network is obtained from the octahedral network by replacing all the octahedra with the icosahedra. An n-dimensional icosahedral network is denoted by ISn. It has 63n2 + 3n number of vertices and 180n2 number of edges. We study the Revan and hyper Revan indices for the icosahedral network as follows.

Theorem 3

Let G = ISn be the Icosahedral network. Then the first and the second Revan indices of ISn are

R1[ISn]=3150n2+150n,R2[ISn]=13950n2+1200n.

Proof

Let G = ISn be the Icosahedral network. Table 2 shows the edge partition of Icosahedral network of ISn based on degrees of end vertices of each edge.

Table 2

Edge partition of Icosahedral Network

(du, dv)Number of edgesrG(u)rG(v)
(5,5)108n2 + 18n1010
(5,10)54n2 — 6n105
(10,10)18n2 — 12n55

Fig. 4

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Fig. 4

Icosahedral network IS3

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

First Revan index of ISn is calculated as

R1[ISn]=uvE[rG(u)+rG(v)]=ueE5,5[rG(u)+rG(v)]+ueE5,10[rG(u)+rG(v)]+ueE10,10[rG(u)+rG(v)]=(108n2+18n)[10+10]+(54n26n)[10+5]+(18n212n)[5+5]=3150n2+150n.

Second Revan index of ISn is calculated as

R2[ISn]=uvE[rG(u)rG(v)]=ueE5,5[rG(u)rG(v)]+ueE5,10[rG(u)rG(v)]+ueE10,10[rG(u)rG(v)]=(108n2+18n)[10×10]+(54n26n)[10×5]+(18n212n)[5×5]=13950n2+1200n.

Theorem 4

Let G = ISn be the icosahedral network. Then the first and the second hyper Revan indices of ISn are

HR1[ISn]=57150n2+4650n,HR2[ISn]=1226250n2+157500n.

Fig. 5

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Fig. 5

The red and blue sheets show the comparison result for Revan index of first and second kind Isocahedral network ISn respectively.

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

Fig. 6

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Fig. 6

The red and blue sheets show the comparison result for hyper Revan index of first and second kind Isocahedral network ISn respectively.

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00004

Proof

Let G = ISn be the icosahedral network. First hyper Revan index of ISn is calculated as

HR1[ISn]=uvE[rG(u)+rG(v)]2=ueE5,5[rG(u)+rG(v)]2+ueE5,10[rG(u)+rG(v)]2+ueE10,10[rG(u)+rG(v)]2=(108n2+18n)[10+10]2+(54n26n)[10+5]2+(18n212n)[5+5]2=57150n2+4650n.

Second hyper Revan index of ISn is calculated as

HR2[ISn]=uvE[rG(u)rG(v)]2=ueE5,5[rG(u)rG(v)]2+ueE5,10[rG(u)rG(v)]2+ueE10,10[rG(u)rG(v)]2=(108n2+18n)[10×10]2+(54n26n)[10×5]2+(18n212n)[5×5]2=1226250n2+157500n.

2 Conclusion

In the present report, we have computed First and second Revan and hyper Revan indices of Octahedral and icosahedral networks.

Communicated by Juan L.G. Guirao

References

  • [1]

    J. Jiang, Y. Zhao, O. M. Yaghi, (2016), Covalent chemistry beyond molecules, Journal of the American Chemical Society, 138(10), 3255–3265.

  • [2]

    Y. Liu et al. (2016), Weaving of organic threads into crystalline covalent organic frameworks, Science, 351, 365–369.

  • [3]

    B. Zhao, J. H. Gan, H. L. Wu, (2016), Redefined Zagreb indices of Some Nano Structures, Applied Mathematics and Nonlinear Sciences, 1, 291–300.

  • [4]

    B. Basavanagoud, W. Gao, S. Patil, V. R. Desai, K. G. Mirajkar, B. Pooja, (2017), Computing first Zagreb index and F-index of new C-products of graphs, Applied Mathematics and Nonlinear Sciences, 2, 285–298.

  • [5]

    A. A. Dobrynin, R. Entringer, I. Gutman, (2001), Wiener index of trees: theory and applications, Acta Applicandae Mathematicae, 66(3), 211–249.

  • [6]

    M. Imran, A. Q. Baig, H. Ali, (2015), On topological properties of dominating David derived networks, Canadian Journal of Chemistry, 94(2), 137–148.

  • [7]

    M. S. Sardar, S. Zafar, M. R. Farahani, (2017), the generalized zagreb index of capra-designed planar benzenoid series cak(c6), Open Journal of Mathematical Sciences, 1(1), 44–51.

  • [8]

    V. R. Kulli, B. Chaluvaraju, H. S. Boregowda, (2017), Connectivity Banhatti indices for certain families of benzenoid systems, Journal of Ultra Chemistry, 13(4), 81–87.

[1]

J. Jiang, Y. Zhao, O. M. Yaghi, (2016), Covalent chemistry beyond molecules, Journal of the American Chemical Society, 138(10), 3255–3265.

[2]

Y. Liu et al. (2016), Weaving of organic threads into crystalline covalent organic frameworks, Science, 351, 365–369.

[3]

B. Zhao, J. H. Gan, H. L. Wu, (2016), Redefined Zagreb indices of Some Nano Structures, Applied Mathematics and Nonlinear Sciences, 1, 291–300.

[4]

B. Basavanagoud, W. Gao, S. Patil, V. R. Desai, K. G. Mirajkar, B. Pooja, (2017), Computing first Zagreb index and F-index of new C-products of graphs, Applied Mathematics and Nonlinear Sciences, 2, 285–298.

[5]

A. A. Dobrynin, R. Entringer, I. Gutman, (2001), Wiener index of trees: theory and applications, Acta Applicandae Mathematicae, 66(3), 211–249.

[6]

M. Imran, A. Q. Baig, H. Ali, (2015), On topological properties of dominating David derived networks, Canadian Journal of Chemistry, 94(2), 137–148.

[7]

M. S. Sardar, S. Zafar, M. R. Farahani, (2017), the generalized zagreb index of capra-designed planar benzenoid series cak(c6), Open Journal of Mathematical Sciences, 1(1), 44–51.

[8]

V. R. Kulli, B. Chaluvaraju, H. S. Boregowda, (2017), Connectivity Banhatti indices for certain families of benzenoid systems, Journal of Ultra Chemistry, 13(4), 81–87.

Journal Information

Figures

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    Octahedral network OT3

  • View in gallery

    The green and blue sheets show the comparison result for Revan index of first and second kind Octahedral network OTn respectively.

  • View in gallery

    The green and blue sheets show the comparison result for hyper Revan index of first and second kind Octahedral network OTn respectively.

  • View in gallery

    Icosahedral network IS3

  • View in gallery

    The red and blue sheets show the comparison result for Revan index of first and second kind Isocahedral network ISn respectively.

  • View in gallery

    The red and blue sheets show the comparison result for hyper Revan index of first and second kind Isocahedral network ISn respectively.

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