Revan Sombor indices: Analytical and statistical study

Abstract

In this paper, we perform analytical and statistical studies of Revan indices on graphs $ G $: $ R(G) = \sum_{uv \in E(G)} F(r_u, r_v) $, where $ uv $ denotes the edge of $ G $ connecting the vertices $ u $ and $ v $, $ r_u $ is the Revan degree of the vertex $ u $, and $ F $ is a function of the Revan vertex degrees. Here, $ r_u = \Delta + \delta - d_u $ with $ \Delta $ and $ \delta $ the maximum and minimum degrees among the vertices of $ G $ and $ d_u $ is the degree of the vertex $ u $. We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $ (a, b) $-$ KA $ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $ (a, b) $-$ KA $ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.