Resumen de Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs

Zoran Stanic

• Let RG0 be the vertex-edge incidence matrix of an oriented graph G. Let Λ(F˙) be the signed graph whose vertices are identified as the edges of a signed graph F, with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of G˙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 F˙ such that R | G R G − 2I is the adjacency matrix of Λ(F˙ ). It occurs that F˙ 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.