On the matching and permanental polynomials of graphs

Abstract

The matching polynomials and permanental polynomials have been investigated extensively, but no investigations focus on the relations between them. In this paper, we establish first two equalities on the signless matching polynomial and signless permanental polynomials. Then we show that the number of perfect matchings of a graph is the mean value of the permanents of skew-adjacency matrices. Moreover, we find that the matching polynomial of a graph is the expectation of skew-permanental polynomials of orientation graphs. Finally, we derive that the variance of the permanents of skew-adjacency matrices of orientation graphs can be expressed in terms of the number of matching alternating cycles in a graph.