On a relationship between the characteristic and matching polynomials of a uniform hypertree

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with characteristic polynomial ${\phi }_T(\lambda )$ and matching polynomial ${\phi }_T(\lambda )$, then ${\phi }_T(\lambda )={\phi }_T(\lambda )\mathrm{.}$ More generally, suppose $\mathcal{T}$ is an $r$-tree of size $m$ with $r\ge 2$. In this paper, we extend the above classical relationship to $r$-trees and establish that $${\phi }_{\mathcal{T}}(\lambda )=\prod _{H⊑\mathcal{T}}{\phi }_H(\lambda {)}^{a_H},$$ where the product is over all connected subgraphs $H$ of $\mathcal{T}$, and the exponent $a_H$ of the factor ${\phi }_H(\lambda )$ can be written as $$a_H=b^{m-e(H)-\mathrm{\mid }\mathrm{\partial }(H)\mathrm{\mid }}{c}^{e(H)}(b-c{)}^{\mathrm{\mid }\mathrm{\partial }(H)\mathrm{\mid }},$$ where $e(H)$ is the size of $H$, $\mathrm{\partial }(H)$ is the boundary of $H$, and $b=(r-1{)}^{r-1},c=r^{r-2}$. In particular, for $r=2$, the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark-Cooper [{\em Electron. J. Combin.}, 2018] and show that for any subgraph $H$ of an $r$-tree $\mathcal{T}$ with $r\ge 3$, ${\phi }_H(\lambda )$ divides ${\phi }_{\mathcal{T}}(\lambda )$, and additionally ${\phi }_H(\lambda )$ divides ${\phi }_{\mathcal{T}}(\lambda )$, if either $r\ge 4$ or $H$ is connected when $r=3$. Moreover, a counterexample is given for the case when $H$ is a disconnected subgraph of a 3-tree.