Topological and spectral properties of random digraphs

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs of size n and connection probability p, D(n,p). In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices V_{x}(D) as well as two vertex-degree-based topological indices: the Randić index R(D) and sum-connectivity index χ(D). First, by performing a scaling analysis, we show that the average degree 〈k〉 serves as a scaling parameter for the average values of V_{x}(D), R(D), and χ(D). Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to (n,p), the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius sqrt[np(1-p)]. Subsequently, we compute six different invariants related to the eigenvalues of D(n,p) and observe that these quantities also scale with sqrt[np(1-p)]. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function (n,p). Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.