ORIENTED UNICYCLIC GRAPHS WITH MINIMAL SKEW RANDIC′ ENERGY

Let G be a simple graph with vertex set V(G) = {v1, v2, . . ., vn}, and G sigma be an orientation of G. Denote by d(vi) the degree of the vertex vi for i = 1, 2, ... , n. The skew Randic ' matrix of G sigma, denoted by RS(G sigma), is the real skew-symmetric matrix (rij)nxn, where / / rij = 1/ d(vi)d(vj) and rji = -1/ d(vi)d(vj) if vi -> vj is an arc of G sigma, otherwise rij = rji = 0. The skew Randic ' energy RES(G sigma) of G sigma is defined as the sum of the norms of all the eigenvalues of RS(G sigma). In this paper, the oriented unicyclic graphs with minimal skew Randic ' energy are determined.