In this work we perform computational and analytical studies of the Randić index R(G) in Erdös-Rényi models G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). First, from a detailed scaling analysis, we show that R(G) = R(G) /(n/2) scales with the product ξ ≈ np, so we can define three regimes: a regime of mostly isolated vertices when ξ < 0.01 (R(G) ≈ 0), a transition regime for 0.01 < ξ < 10 (where 0 < R(G) < n/2), and a regime of almost complete graphs for ξ > 10 (R(G) ≈ n/2). Then, motivated by the scaling of R(G) , we analytically (i) obtain new relations connecting R(G) with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on R(G) for graphs in Erdös-Rényi models.

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and n − m vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization-to-delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.

We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős–Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α , and the losses-and-gain strength γ . Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude i γ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ ≡ ξ ( N , α , γ ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ < 0.1 ( 10 < ξ ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1 < ξ < 10 . Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ , the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.