Skew-spectra and skew energy of various products of graphs

Document Type: Research Paper

Authors

1 Center for Combinatorics and LPMC-TJKLC, Nankai University

2 College of Science, China University of Mining and Technology

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 optimum 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 $G$‎.

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