Revan-degree indices on random graphs

Given a simple connected non-directed graph $G=(V(G),E(G))$, we consider two families of graph invariants: $RX_\Sigma(G) = \sum_{uv \in E(G)} F(r_u,r_v)$ (which has gained interest recently) and $RX_\Pi(G) = \prod_{uv \in E(G)} F(r_u,r_v)$ (that we introduce in this work); 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 $u$. Particularly, we apply both $RX_\Sigma(G)$ and R$X_\Pi(G)$ on two models of random graphs: Erd\"os-R\'enyi graphs and random geometric graphs. By a thorough computational study we show that \left and \left, normalized to the order of the graph, scale with the average Revan degree \left; here \left denotes the average over an ensemble of random graphs. Moreover, we provide analytical expressions for several graph invariants of both families in the dense graph limit.Comment: 16 pages, 10 figure

Spectral and localization properties of random bipartite graphs

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 $\alpha\in[0,1]$ 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 $\xi\equiv\xi(n,m,\alpha)$ that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when $\xi10$) the eigenvectors are localized (extended), whereas the localization--to--delocalization transition occurs in the interval $1/10<\xi<10$. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed $\xi$, the spectral properties of our graph model are also universal.Comment: 17 pages, 10 figure