Hermite polynomials and a duality relation for matchings polynomials

Abstract

LetG be a graph onn vertices. Ak-matching inG is a set ofk independent edges. If 2k=n then ak-matching is called perfect. The number ofk-matchings inG isp(G, k). (We setp(G, 0)=1). The matchings polynomial ofG is

$$\alpha (G,x) = \sum\limits_{k = 0}^{[n/2]} {( - 1)^k p(G,k)x^{n - 2k} } $$

Our main result is that the number of perfect matchings in the complement ofG is equal to

$$(2\pi )^{ - 1/2} \int\limits_{ - \infty }^\infty {\alpha (G,x)} \exp ( - x^2 /2)dx.$$

((1))

LetK _m be the complete graph onm vertices. Then α(K _m ,x) is the Hermite polynomial He _n (x) of degreen. Using (1) we show, amongst other results, that

$$\alpha (\bar G,x) = \sum\limits_{k = 0}^{[n/2]} {p(G,k)} \alpha (K_{n - 2k} ,x).$$