Revan-degree indices on random graphs

Abstract

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 $\left< RX_\Sigma(G) \right>$ and $\left< \ln RX_\Pi(G) \right>$, normalized to the order of the graph, scale with the average Revan degree $\left< r \right>$; here $\left< \cdot \right>$ 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.