The skew energy of random oriented graphs

TL;DR

This paper investigates the skew energy of random oriented graphs, providing precise estimates for almost all such graphs by extending Wigner's semicircle law, including for regular cases.

Contribution

It generalizes Wigner's semicircle law to estimate the skew energy of random and regular oriented graphs with exact formulas.

Findings

Derived exact estimates for skew energy of random oriented graphs

Extended semicircle law to oriented graph spectra

Provided estimates for regular oriented graphs

Abstract

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the absolute values of all the eigenvalues of $S(G^\sigma)$. In this paper, we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing Wigner's semicircle law. Moreover, we consider the skew energy of random regular oriented graphs $G_{n,d}^\sigma$, and get an exact estimate of the skew energy for almost all regular oriented graphs.