On the skew eigenvalues of joined union of oriented graphs and applications

Abstract

Let be an oriented graph with n vertices and m arcs having underlying graph G. The skew matrix of oriented graph , denoted by is a (−1, 0, 1)- skew symmetric matrix. The skew eigenvalues of are the eigenvalues of the matrix and its characteristic polynomial is the skew characteristic polynomial of . The sum of the absolute values of the skew eigenvalues is the skew energy of and is denoted by . In this paper, we extend the definition of joined union of graphs to oriented graphs. We show that the skew eigenvalues of the joined union of oriented graphs is the union of the skew eigenvalues of the component oriented graphs except some eigenvalues, which are given by an auxiliary matrix associated with the joined union. As a special case we obtain the skew eigenvalues of join of two oriented graphs and the lexicographic product of oriented graphs. We provide examples of orientations of some well known graphs to highlight the importance of our results. As applications to our result we obtain some new infinite families of skew equienergetic oriented graphs. Our results extend and generalize the results obtained in [H.S. Ramane, K.C. Nandeesh, I. Gutman and X. Li, Skew equienergetic digraphs, Trans. Comb., 5(1), (2016) 15-23].