On the skew characteristics polynomial/eigenvalues of operations on bipartite oriented graphs and applications

Let G→ be an oriented graph with n vertices and m arcs having underlying graph G. The skew matrix of G→, denoted by S(G→) is a (−1,0,1) -skew symmetric matrix. The skew eigenvalues of G→ are the eigenvalues of S(G→) and its characteristic polynomial is the skew characteristic polynomial of G→. The sum of the absolute values of the skew eigenvalues is the skew energy of G→ and is denoted by ES(G→). In this paper, we study the skew characteristic polynomial and skew eigenvalues of joined union of oriented bipartite graphs and some of its variations. We show that the skew eigenvalues of the joined union of oriented bipartite graphs and some variations of oriented bipartite 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 bipartite graphs and the lexicographic product of an oriented graph and an oriented bipartite graph. Some examples of orientations of well-known graphs are presented to highlight the importance of the results. As applications to our result we obtain some new infinite families of skew equienergetic oriented graphs. Our results extend and generalize some of the results obtained in [C. Adiga and B.R. Rakshith, More skew-equienergetic digraphs, Commun. Comb. Optim., 1(1) (2016) 55–71].