Analytical and statistical studies of Rodriguez–Velazquez indices

Abstract

In this work we perform analytical and statistical studies of the Rodriguez–Velazquez (RV) indices on graphs G. The topological RV(G) indices, recently introduced in Rodriguez–Velazquez and Balaban (J Math Chem 57:1053, 2019), are based on graph adjacency matrix eigenvalues and eigenvectors. First, we analytically obtain new relations connecting RV(G) with the graph energy E(G) and the subgraph centrality EE(G), the later being proportional to the well known Estrada index. Then, within a random matrix theory (RMT) approach we statistically validate our relations on ensembles of randomly-weighted Erdős–Renyi graphs G(n, p), characterized by n vertices connected independently with probability $$p \in (0,1)$$ . Additionally, we show that the ratio $$\left\langle RV(G(n,p)) \right\rangle /\left\langle RV(G(n,0)) \right\rangle$$ scales with the average degree $$\left\langle k \right\rangle =(n-1)p$$ .