4-Regular oriented graphs with optimum skew energy

TL;DR

This paper characterizes 4-regular graphs with orientations that maximize skew energy, providing explicit orientations and revealing infinitely many such optimal graphs, contrasting with the limited 3-regular case.

Contribution

It identifies all underlying graphs of 4-regular oriented graphs with maximum skew energy and constructs orientations achieving this optimum.

Findings

Characterization of underlying graphs for 4-regular optimum skew energy graphs

Explicit orientations that attain maximum skew energy

Existence of infinitely many such optimal graphs for 4-regular case

Abstract

Let $G$ be a simple undirected graph, and $G^\sigma$ 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 characterize the underlying graphs of all 4-regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum. It should be pointed out that there are infinitely many 4-regular connected optimum skew energy oriented graphs, while the 3-regular case only has two graphs: $K_4$ the complete graph on 4 vertices and $Q_3$ the hypercube.