Topological and spectral properties of random digraphs

Abstract

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs $D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of non-isolated vertices $V_x(D)$ as well as two vertex-degree-based topological indices: the Randić index $R(D)$ and sum-connectivity index $\chi (D)$. First, by performing a scaling analysis we show that the average degree $⟨k⟩$ serves as scaling parameter for the average values of $V_x(D)$, $R(D)$ and $\chi (D)$. Then, we also state expressions relating the number of arcs, spectral radius, and closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs. Concerning spectral properties, we compute six different graph energies on $D(n,p)$. We start by validating $⟨k⟩$ as the scaling parameter of the graph energies. Additionally, we reformulate a set of bounds previously reported in the literature for these energies as a function $(n,p)$. Finally, we phenomenologically state relations between energies that allow us to extend previously known bounds.