stylistic:style_transfer, structural:expansion
Connections to other polynomials

The first type of matching polynomial is a direct generalization of the rook polynomial.

The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = K, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial L (x) by the identity:

If G is the complete graph K, then M (x) is an Hermite polynomial:

where H (x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by Godsil (1981).

If G is a forest, then its matching polynomial is equal to the characteristic polynomial of its adjacency matrix.

If G is a path or a cycle, then M (x) is a Chebyshev polynomial. In this caseμ (1, x) is a Fibonacci polynomial or Lucas polynomial respectively.