Mehler formulae for matching polynomials of graphs and independence polynomials of clawfree graphs

Abstract

The independence polynomial of a graph G is the polynomial ∑Ix|I|, summed over all independent subsets I⊆V(G). We show that if G is clawfree, then there exists a Mehler formula for its independence polynomial. This was proved for matching polynomials in Lass (2004) [19] and extends the combinatorial proof of the Mehler formula found by Foata (1978) [9]. It implies immediately that all the roots of the independence polynomial of a clawfree graph are real, answering a question posed by Hamidoune (1990) [14] and Stanley (1998) [28] and solved by Chudnovsky and Seymour (2007) [6]. We also prove a Mehler formula for the multivariate matching polynomial, extending results of Lass (2004) [19].