Skew-spectra and skew energy of various products of graphs

TL;DR

This paper investigates the skew spectra and skew energy of various graph products, providing new orientations and constructing families of graphs with maximum skew energy, advancing understanding in spectral graph theory.

Contribution

It introduces orientations for the Kronecker, strong, and lexicographic products of graphs and determines their skew spectra, leading to new maximum skew energy graph families.

Findings

Determined skew spectra for Kronecker and strong products of graphs.

Constructed new families of graphs with maximum skew energy.

Analyzed skew energy of lexicographic product orientations.

Abstract

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ consisting of all the eigenvalues of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$. In this paper, we give orientations of the Kronecker product $H\otimes G$ and the strong product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an arbitrary graph. Then we determine the skew-spectra of the resultant oriented graphs. As applications, we construct new families of oriented graphs with maximum skew energy. Moreover, we consider the skew energy of the orientation of the lexicographic product $H[G]$ of a bipartite graph $H$ and a graph $G$.