SOME RELATIONS BETWEEN THE SKEW SPECTRUM OF AN ORIENTED GRAPH AND THE SPECTRUM OF CERTAIN CLOSELY ASSOCIATED SIGNED GRAPHS

Let R-G' be the vertex-edge incidence matrix of an oriented graph G'. Let (F (over dot)) be the signed graph whose vertices are identified as the edges of a signed graph F (over dot), with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of G (over dot) are adjacent and have the same (resp. different) sign. In this paper, we prove that G' is bipartite if and only if there exists a signed graph (over dot) such that R-G'R-T(G') - 2I is the adjacency matrix of lambda(F (over dot)). It occurs that F (over dot) is fully determined by G'. As an application, in some particular cases we express the skew eigenvalues of G' in terms of the eigenvalues of F. We also establish some upper bounds for the skew spectral radius of G' in both the bipartite and the non-bipartite case.y