On the General Randi\'c index of polymeric networks modelled by generalized Sierpi\'nski graphs

TL;DR

This paper derives formulas for the general Randić index of Sierpiński-type polymeric networks with arbitrary base graphs, extending previous work that focused on specific base graphs like complete or bipartite graphs.

Contribution

It provides closed-form expressions for the Randić index $R_{\alpha}$ for a broad class of Sierpiński-type networks with any base graph, generalizing earlier results.

Findings

Closed formulas for $R_{\alpha}$ of Sierpiński networks with arbitrary base graphs.

Extension of previous results from specific base graphs to general cases.

Applicable to various graph types, broadening the understanding of Randić indices in complex networks.

Abstract

The General Randi\'c index $R_\alpha$ of a simple graph $G$ is defined as \[ R_\alpha(G)=\sum_{v_{i}\sim v_{j}} (\delta_{i}\delta_{j})^\alpha, \] where $\delta_i$ denotes the degree of the vertex $v_i$. Rodr\'iguez-Vel\'azquez and Tom\'as-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145--160] obtained closed formulae for the Randi\'c index $R_{-1/2}$ of Sierpi\'nski-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randi\'c index $R_{\alpha}$ of Sierpi\'nski-type polymeric networks, where the base graph is arbitrary.