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

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.