Revan and hyper-Revan indices of 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 R 1 ( G ) = ∑ u v ∈ E $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$ [ r G ( u ) + r G ( v )] and R 2 ( G ) = ∑ u v ∈ E $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$ [ r G ( u ) r G ( 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 HR 1 ( G ) = ∑ u v ∈ E $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$ [ r G ( u ) + r G ( v )] 2 and HR 2 ( G ) = ∑ u v ∈ E $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$ [ r G ( u ) r G ( v )] 2 . In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.