A survey on the skew energy of oriented graphs

TL;DR

This survey reviews the main results and open problems related to the skew energy of oriented graphs, including its variants like skew Laplacian and Randić energies, highlighting its mathematical properties and applications.

Contribution

It provides a comprehensive summary of existing research on skew energy in oriented graphs and introduces open problems for future investigation.

Findings

Summarizes key results on skew energy of oriented graphs

Discusses variants like skew Laplacian and Randić energies

Proposes open problems for further research

Abstract

Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical applications, it causes great concern and has many generalizations. The skew energy and skew energy-like are the generalizations in oriented graphs. Let $G^\sigma$ be an oriented graph of $G$ with skew adjacency matrix $S(G^\sigma)$. The skew energy of $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all eigenvalues of $S(G^\sigma)$, which was introduced by Adiga, Balakrishnan and So in 2010. In this paper, we summarize main results on the skew energy of oriented graphs. Some open problems are proposed for further study. Besides, results on the skew energy-like: the skew Laplacian energy and skew Randi\'{c} energy are also surveyed at the end.