Revan and hyper-Revan indices of Octahedral and icosahedral networks

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 r_G (v) = Δ(G) + δ(G) – d_G(v). The first and second Revan indices of G are defined as:

$\begin{matrix} R_{1} (G) = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)], \end{matrix}$ $$\begin{array}{} \displaystyle R_{1}(G)=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)], \end{array}$$

and

$\begin{matrix} R_{2} (G) = \sum_{u v \in E} [r_{G} (u) r_{G} (v)], \end{matrix}$ $$\begin{array}{} \displaystyle R_{2}(G)=\sum\limits_{uv\in E}[r_{G}(u)r_{G}(v)], \end{array}$$

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

$\begin{matrix} H R_{1} (G) = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)]^{2} \end{matrix}$ $$\begin{array}{} \displaystyle HR_{1}(G)=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)]^{2} \end{array}$$

$\begin{matrix} H R_{2} (G) = \sum_{u v \in E} [r_{G} (u) r_{G} (v)]^{2} \end{matrix}$ $$\begin{array}{} \displaystyle HR_{2}(G)=\sum\limits_{uv\in E}[r_{G}(u)r_{G}(v)]^{2} \end{array}$$

where r_G (v) = Δ(G) + δ(G) –d_G (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 OT_n, where n is the order of circumscribing. The numbers of vertices and edges in OT_n with n ≥ 1 are 27n² + 3n and 72n² respectively. We study the Revan indices for the octahedral network as follows.

Theorem 1

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

$\begin{matrix} R_{1} [O T_{n}] = 864 n^{2} + 96 n, \\ R_{2} [O T_{n}] = 2592 n^{2} + 576 n . \end{matrix}$ $$\begin{array}{} \displaystyle R_{1}[OT_n]=864{n}^{2}+ 96n,\\ \displaystyle R_{2}[OT_n]=2592{n}^{2}+ 576n. \end{array}$$

Proof

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

$\begin{matrix} R_{1} [O T_{n}] = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)] \\ = \sum_{u v \in E_{4, 4}} [r_{G} (u) + r_{G} (v)] + \sum_{u v \in E_{4, 8}} [r_{G} (u) + r_{G} (v)] \\ + \sum_{u v \in E_{8, 8}} [r_{G} (u) + r_{G} (v)] \\ = (18 n^{2} + 12 n) [8 + 8] + 36 n^{2} [8 + 4] + (18 n^{2} - 12 n) [4 + 4] \\ = 864 n^{2} + 96 n \end{matrix}$ $$\begin{array}{} \displaystyle R_{1}[OT_n]=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\quad~~=\sum\limits_{uv\in E_{4,4}}[r_{G}(u)+r_{G}(v)]+\sum\limits_{uv\in E_{4,8}}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\qquad\quad+\sum\limits_{uv\in E_{8,8}}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\quad~~= (18n^{2}+12n)[8+8]+36n^{2}[8+4]+(18n^{2}-12n)[4+4]\\ \displaystyle \qquad\quad~~=864{n}^{2}+ 96n \end{array}$$

Table 1

Edge partition of octahedral network

(d_u, dv)	Number of edges	r_G(u)	r_G(v)
(4,4)	18n² + 12n	8	8
(4,8)	36n²	8	4
(8,8)	18n² — 12n	4	4

Second Revan index of OT_n is calculated as

$\begin{matrix} R_{2} [O T_{n}] = \sum_{u e} [r_{G} (u) r_{G} (v)] \\ = \sum_{u v \in E_{4, 4}} [r_{G} (u) r_{G} (v)] + \sum_{u v \in E_{4, 8}} [r_{G} (u) r_{G} (v)] \\ + \sum_{u v \in E_{8, 8}} [r_{G} (u) r_{G} (v)] \\ = (18 n^{2} + 12 n) [8 \times 8] + 36 n^{2} [8 \times 4] + (18 n^{2} - 12 n) [4 \times 4] \\ = 2592 n^{2} + 576 n . \end{matrix}$ $$\begin{array}{} \displaystyle R_{2}[OT_n]=\sum\limits_{ue}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\quad~~=\sum\limits_{uv\in E_{4,4}}[r_{G}(u)r_{G}(v)]+\sum\limits_{uv\in E_{4,8}}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\qquad \quad+\sum\limits_{uv\in E_{8,8}}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\quad~~= (18n^{2}+12n)[8\times8]+36n^{2}[8\times4]+(18n^{2}-12n)[4\times4]\\ \displaystyle \qquad\quad~~= 2592{n}^{2}+ 576n. \end{array}$$

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

Theorem 2

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

$\begin{matrix} H R_{1} [O T_{n}] = 10944 n^{2} + 2304 n, \\ H R_{2} [O T_{n}] = 115200 n^{2} + 46080 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{1}[OT_n]=10944{n}^{2}+ 2304n,\\ \displaystyle HR_{2}[OT_n]=115200{n}^{2}+ 46080n. \end{array}$$

Proof

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

$\begin{matrix} H R_{1} [O T_{n}] = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)]^{2} \\ = \sum_{u v \in E_{4, 4}} [r_{G} (u) + r_{G} (v)]^{2} + \sum_{u v \in E_{4, 8}} [r_{G} (u) + r_{G} (v)]^{2} \\ + \sum_{u v \in E_{8, 8}} [r_{G} (u) + r_{G} (v)]^{2} \\ = (18 n^{2} + 12 n) [8 + 8]^{2} + 36 n^{2} [8 + 4]^{2} + (18 n^{2} - 12 n) [4 + 4]^{2} \\ = 10944 n^{2} + 2304 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{1}[OT_n]=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)]^2\\ \displaystyle \qquad\qquad~~=\sum\limits_{uv\in E_{4,4}}[r_{G}(u)+r_{G}(v)]^2+\sum\limits_{uv\in E_{4,8}}[r_{G}(u)+r_{G}(v)]^2\\ \displaystyle \qquad\qquad\qquad+\sum\limits_{uv\in E_{8,8}}[r_{G}(u)+r_{G}(v)]^2\\ \displaystyle \qquad\qquad~~= (18n^{2}+12n)[8+8]^2+36n^{2}[8+4]^2+(18n^{2}-12n)[4+4]^2\\ \displaystyle \qquad\qquad~~=10944{n}^{2}+ 2304n. \end{array}$$

Second hyper Revan index of OT_n is calculated as

$\begin{matrix} H R_{2} [O T_{n}] = \sum_{u v} [r_{G} (u) r_{G} (v)]^{2} \\ = \sum_{u v \in E_{4, 4}} [r_{G} (u) r_{G} (v)]^{2} + \sum_{u v \in E_{4, 8}} [r_{G} (u) r_{G} (v)]^{2} \\ + \sum_{u v \in E_{8, 8}} [r_{G} (u) r_{G} (v)]^{2} \\ = (18 n^{2} + 12 n) [8 \times 8]^{2} + 36 n^{2} [8 \times 4]^{2} + (18 n^{2} - 12 n) [4 \times 4]^{2} \\ = 115200 n^{2} + 46080 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{2}[OT_n]=\sum\limits_{uv}[r_{G}(u)r_{G}(v)]^{2}\\ \displaystyle \qquad\qquad~~=\sum\limits_{uv\in E_{4,4}}[r_{G}(u)r_{G}(v)]^2+\sum\limits_{uv\in E_{4,8}}[r_{G}(u)r_{G}(v)]^2\\ \displaystyle \qquad\qquad \qquad+\sum\limits_{uv\in E_{8,8}}[r_{G}(u)r_{G}(v)]^2\\ \displaystyle \qquad\qquad~~= (18n^{2}+12n)[8\times8]^2+36n^{2}[8\times4]^2+(18n^{2}-12n)[4\times4]^2\\ \displaystyle \qquad\qquad~~=115200{n}^{2}+ 46080n. \end{array}$$

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

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 IS_n. 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 = IS_n be the Icosahedral network. Then the first and the second Revan indices of IS_n are

$\begin{matrix} R_{1} [I S_{n}] & = & 3150 n^{2} + 150 n, \\ R_{2} [I S_{n}] & = & 13950 n^{2} + 1200 n . \end{matrix}$ $$\begin{array}{} \displaystyle R_{1}[IS_n]&=& 3150{n}^{2}+ 150n,\\ \displaystyle R_{2}[IS_n]&=& 13950{n}^{2}+ 1200n. \end{array}$$

Proof

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

Table 2

Edge partition of Icosahedral Network

(d_u, dv)	Number of edges	r_G(u)	r_G(v)
(5,5)	108n² + 18n	10	10
(5,10)	54n² — 6n	10	5
(10,10)	18n² — 12n	5	5

First Revan index of IS_n is calculated as

$\begin{matrix} R_{1} [I S_{n}] = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)] \\ = \sum_{u e \in E_{5, 5}} [r_{G} (u) + r_{G} (v)] + \sum_{u e \in E_{5, 10}} [r_{G} (u) + r_{G} (v)] \\ + \sum_{u e \in E_{10, 10}} [r_{G} (u) + r_{G} (v)] \\ = (108 n^{2} + 18 n) [10 + 10] + (54 n^{2} - 6 n) [10 + 5] \\ + (18 n^{2} - 12 n) [5 + 5] \\ = 3150 n^{2} + 150 n . \end{matrix}$ $$\begin{array}{} \displaystyle R_{1}[IS_n]=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\quad~=\sum\limits_{ue\in E_{5,5}}[r_{G}(u)+r_{G}(v)]+ \sum\limits_{ue\in E_{5,10}}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\qquad~~+\sum\limits_{ue\in E_{10,10}}[r_{G}(u)+r_{G}(v)]\\ \displaystyle \qquad\quad~=(108n^2+18n)[10+10]+(54n^{2}-6n)[10+5]\\ \displaystyle \qquad\qquad~+(18n^2-12n)[5+5]\\ \displaystyle \qquad\quad~=3150{n}^{2}+ 150n. \end{array}$$

Second Revan index of IS_n is calculated as

$\begin{matrix} R_{2} [I S_{n}] = \sum_{u v \in E} [r_{G} (u) r_{G} (v)] \\ = \sum_{u e \in E_{5, 5}} [r_{G} (u) r_{G} (v)] + \sum_{u e \in E_{5, 10}} [r_{G} (u) r_{G} (v)] \\ + \sum_{u e \in E_{10, 10}} [r_{G} (u) r_{G} (v)] \\ = (108 n^{2} + 18 n) [10 \times 10] + (54 n^{2} - 6 n) [10 \times 5] \\ + (18 n^{2} - 12 n) [5 \times 5] \\ = 13950 n^{2} + 1200 n . \end{matrix}$ $$\begin{array}{} \displaystyle R_{2}[IS_n]=\sum\limits_{uv\in E}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\quad~=\sum\limits_{ue\in E_{5,5}}[r_{G}(u)r_{G}(v)]+ \sum\limits_{ue\in E_{5,10}}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\qquad~~+\sum\limits_{ue\in E_{10,10}}[r_{G}(u)r_{G}(v)]\\ \displaystyle \qquad\quad~=(108n^2+18n)[10\times10]+(54n^{2}-6n)[10\times5]\\ \displaystyle \qquad\qquad~~+(18n^2-12n)[5\times5]\\ \displaystyle \qquad\quad~=13950{n}^{2}+ 1200n. \end{array}$$

Theorem 4

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

$\begin{matrix} H R_{1} [I S_{n}] = 57150 n^{2} + 4650 n, \\ H R_{2} [I S_{n}] = 1226250 n^{2} + 157500 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{1}[IS_n]=57150{n}^{2}+ 4650n,\\ \displaystyle HR_{2}[IS_n]=1226250{n}^{2}+ 157500n. \end{array}$$

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

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

Proof

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

$\begin{matrix} H R_{1} [I S_{n}] = \sum_{u v \in E} [r_{G} (u) + r_{G} (v)]^{2} \\ = \sum_{u e \in E_{5, 5}} [r_{G} (u) + r_{G} (v)]^{2} + \sum_{u e \in E_{5, 10}} [r_{G} (u) + r_{G} (v)]^{2} \\ + \sum_{u e \in E_{10, 10}} [r_{G} (u) + r_{G} (v)]^{2} \\ = (108 n^{2} + 18 n) [10 + 10]^{2} + (54 n^{2} - 6 n) [10 + 5]^{2} \\ + (18 n^{2} - 12 n) [5 + 5]^{2} \\ = 57150 n^{2} + 4650 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{1}[IS_n]=\sum\limits_{uv\in E}[r_{G}(u)+r_{G}(v)]^{2}\\ \displaystyle \qquad\qquad~=\sum\limits_{ue\in E_{5,5}}[r_{G}(u)+r_{G}(v)]^2+ \sum\limits_{ue\in E_{5,10}}[r_{G}(u)+r_{G}(v)]^2\\ \displaystyle \qquad\qquad\quad~~+\sum\limits_{ue\in E_{10,10}}[r_{G}(u)+r_{G}(v)]^2\\ \displaystyle \qquad\qquad~=(108n^2+18n)[10+10]^2+(54n^{2}-6n)[10+5]^2\\ \displaystyle \qquad\qquad\quad~~+(18n^2-12n)[5+5]^2\\ \displaystyle \qquad\qquad~=57150{n}^{2}+ 4650n. \end{array}$$

Second hyper Revan index of IS_n is calculated as

$\begin{matrix} H R_{2} [I S_{n}] = \sum_{u v \in E} [r_{G} (u) r_{G} (v)]^{2} \\ = \sum_{u e \in E_{5, 5}} [r_{G} (u) r_{G} (v)]^{2} + \sum_{u e \in E_{5, 10}} [r_{G} (u) r_{G} (v)]^{2} \\ + \sum_{u e \in E_{10, 10}} [r_{G} (u) r_{G} (v)]^{2} \\ = (108 n^{2} + 18 n) [10 \times 10]^{2} + (54 n^{2} - 6 n) [10 \times 5]^{2} \\ + (18 n^{2} - 12 n) [5 \times 5]^{2} \\ = 1226250 n^{2} + 157500 n . \end{matrix}$ $$\begin{array}{} \displaystyle HR_{2}[IS_n]=\sum\limits_{uv\in E}[r_{G}(u)r_{G}(v)]^{2}\\ \displaystyle \qquad\qquad~=\sum\limits_{ue\in E_{5,5}}[r_{G}(u)r_{G}(v)]^2+ \sum\limits_{ue\in E_{5,10}}[r_{G}(u)r_{G}(v)]^2\\ \displaystyle \qquad\qquad\quad~~+\sum\limits_{ue\in E_{10,10}}[r_{G}(u)r_{G}(v)]^2\\ \displaystyle \qquad\qquad~=(108n^2+18n)[10\times10]^2+(54n^{2}-6n)[10\times5]^2\\ \displaystyle \qquad\qquad\quad~~+(18n^2-12n)[5\times5]^2\\ \displaystyle \qquad\qquad~=1226250{n}^{2}+ 157500n. \end{array}$$

Conclusion

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

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Revan and hyper-Revan indices of Octahedral and icosahedral networks

Published Online: Oct 03, 2018

Page range: 33 - 40

Received: Nov 07, 2017

Accepted: Feb 27, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00004

Keywords
Revan index, hyper-Revan, Octahedral, Icosahedral, Networks

© 2018 A. Q. Baig et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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Revan and hyper-Revan indices of Octahedral and icosahedral networks

Published Online: Oct 03, 2018

Page range: 33 - 40

Received: Nov 07, 2017

Accepted: Feb 27, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00004

KeywordsRevan index, hyper-Revan, Octahedral, Icosahedral, Networks

© 2018 A. Q. Baig et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Fig. 2

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Keywords
Revan index, hyper-Revan, Octahedral, Icosahedral, Networks