The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let GσGσ be a weighted oriented graph with skew adjacency matrix S(Gσ)S(Gσ). Then GσGσ is usually referred as the weighted oriented graph associated to S(Gσ)S(Gσ). Denote by ϕ(Gσ;λ)ϕ(Gσ;λ) the characteristic polynomial of the weighted oriented graph GσGσ, which is defined asϕ(Gσ;λ)=det(λIn-S(Gσ))=∑i=0nai(Gσ)λn-i.In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.