<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>G</italic> be a connected graph with vertex set <italic>V</italic>(<italic>G</italic>) and edge set <italic>E</italic>(<italic>G</italic>). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of <italic>G</italic> are defined as <italic>R</italic>1(<italic>G</italic>) = <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mstyle><munder><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></munder></mstyle></mtd></mtr></mtable></math><tex-math>$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$</tex-math></alternatives></inline-formula>[<italic>rG</italic>(<italic>u</italic>) + <italic>rG</italic>(<italic>v</italic>)] and <italic>R</italic>2(<italic>G</italic>) = <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mstyle><munder><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></munder></mstyle></mtd></mtr></mtable></math><tex-math>$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$</tex-math></alternatives></inline-formula>[<italic>rG</italic>(<italic>u</italic>)<italic>rG</italic>(<italic>v</italic>)], where <italic>uv</italic> means that the vertex <italic>u</italic> and edge <italic>v</italic> are adjacent in <italic>G</italic>. The first and second hyper-Revan indices of <italic>G</italic> are defined as <italic>HR</italic>1(<italic>G</italic>) = <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mstyle><munder><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></munder></mstyle></mtd></mtr></mtable></math><tex-math>$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$</tex-math></alternatives></inline-formula>[<italic>rG</italic>(<italic>u</italic>) + <italic>rG</italic>(<italic>v</italic>)]2 and <italic>HR</italic>2(<italic>G</italic>) = <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mstyle><munder><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></munder></mstyle></mtd></mtr></mtable></math><tex-math>$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$</tex-math></alternatives></inline-formula>[<italic>rG</italic>(<italic>u</italic>)<italic>rG</italic>(<italic>v</italic>)]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) = ∑uv∈E$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$[rG(u) + rG(v)] and R2(G) = ∑uv∈E$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$[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) = ∑uv∈E$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$[rG(u) + rG(v)]2 and HR2(G) = ∑uv∈E$\begin{array}{}
\displaystyle
\sum\limits_{uv\in E}
\end{array}$[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.

Revan and hyper-Revan indices of Octahedral and icosahedral networks

Applied Mathematics and Nonlinear Sciences

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

{"article-title":"Revan and hyper-Revan indices of Octahedral and icosahedral networks"}

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...

(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

(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

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.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Edge partition of octahedral network

Edge partition of Icosahedral Network

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

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Edge partition of octahedral network

Edge partition of Icosahedral Network

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