A new expression for matching polynomials

Let GG be an arbitrary simple graph. Godsil and Gutman in 1978 and Yan et al. in 2005 established different expressions for the matching polynomial μ(G,x)μ(G,x) in terms of det(xIn−H)det(xIn−H) for some families of matrices HH. This paper improves their results and simplifies the computation of μ(G,x)μ(G,x).