On oriented graphs with minimal skew energy

TL;DR

This paper derives an integral formula for the skew energy of oriented graphs and identifies those with minimal skew energy among connected graphs with specified vertices and arcs, extending concepts from undirected graph energy.

Contribution

It introduces a new integral formula for skew energy and characterizes all minimal skew energy graphs within certain parameters, advancing the understanding of skew energy in oriented graphs.

Findings

Derived an integral formula for skew energy.

Identified all connected oriented graphs with minimal skew energy for given vertices and arcs.

Extended the conjecture on graph energy to skew energy in oriented graphs.

Abstract

Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m \ (n\le m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi$\acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]