5-regular oriented graphs with optimum skew energy

TL;DR

This paper characterizes all connected 5-regular oriented graphs that achieve the maximum possible skew energy, extending understanding of energy bounds in graph theory.

Contribution

It identifies all connected 5-regular oriented graphs with maximum skew energy, providing a complete classification in this specific case.

Findings

All connected 5-regular oriented graphs with maximum skew energy are determined.

The maximum skew energy for these graphs is explicitly characterized.

The results extend previous bounds on skew energy for regular graphs.

Abstract

Let $G$ be a simple undirected graph and $G^\sigma$ be the corresponding oriented graph of $G$ with the orientation $\sigma$. The skew energy of $G^\sigma$, denoted by $\varepsilon_s(G^\sigma)$, is defined as the sum of the singular values of the skew adjacency matrix $S(G^\sigma)$. In 2010, Adiga et al. certified that $\varepsilon_s(G^\sigma) \leq n\sqrt{\Delta}$, where $\Delta$ is the maximum degree of $G$ of order $n$. In this paper, we determine all connected 5-regular oriented graphs of order $n$ with maximum skew-energy.