Distribution of zeros of matching polynomials of hypergraphs

Abstract

Let \h be a connected $k$-graph with maximum degree ${\Delta}\geq 2$ and let \mu(\h, x) be the matching polynomial of \h. In this paper, we focus on studying the distribution of zeros of the matching polynomials of $k$-graphs. We prove that the zeros (with multiplicities) of \mu(\h, x) are invariant under a rotation of an angle $2\pi/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. Let \lambda(\h) denote the maximum modulus of all zeros of \mu(\h, x). We show that \lambda(\h) is a simple root of \mu(\h, x) and \Delta^{1\over k} \leq \lambda(\h)< \frac{k}{k-1}\big((k-1)(\Delta-1)\big)^{1\over k}. To achieve these, we introduce the path tree \T(\h,u) of \h with respect to a vertex $u$ of \h, which is a $k$-tree, and prove that \frac{\mu(\h-u,x)}{\mu(\h, x)} = \frac{\mu(\T(\h,u)-u,x) }{\mu(\T(\h,u),x)}, which generalizes the celebrated Godsil's identity on the matching polynomial of graphs