Spectra of weighted uniform hypertrees

Abstract

Let $T$ be a $k$-tree equipped with a weighting function $\w: V(T)\cup E(T)\rightarrow \C$, where $k \geq 3$. The weighted matching polynomial of the weighted $k$-tree $(T,\w)$ is defined to be $$ \mu(T,\w,x)= \sum_{M \in \mathcal{M}(T)}(-1)^{|M|}\prod_{e \in E(M)}\mathbf{w}(e)^k \prod_{v \in V(T)\backslash V(M)}(x-\w(v)), $$ where $\mathcal{M}(T)$ denotes the set of matchings (including empty set) of $T$. In this paper, we investigate the eigenvalues of the adjacency tensor $\A(T,\w)$ of the weighted $k$-tree $(T,\w)$. The main result provides that $\w(v)$ is an eigenvalue of $\A(T,\w)$ for every $v\in V(T)$, and if $\lambda\neq \w(v)$ for every $v\in V(T)$, then $\lambda$ is an eigenvalue of $\A(T,\w)$ if and only if there exists a subtree $T'$ of $T$ such that $\lambda$ is a root of $\mu(T',\w,x)$. Moreover, the spectral radius of $\A(T,\w)$ is equal to the largest root of $\mu(T,\w,x)$ when $\w$ is real and nonnegative. The result extends a work by Clark and Cooper ({\em On the adjacency spectra of hypertrees, Electron. J. Combin., 25 (2)(2018) $\#$P2.48}) to weighted $k$-trees. As applications, two analogues of the above work for the Laplacian and the signless Laplacian tensors of $k$-trees are obtained.