Distribution of zeros of matching polynomials of hypergraphs

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

Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.Comment: This research was supported by NSF Grant CCF-155375, 20 page