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 $\mathcal{A}(T,w)$ of the weighted $k$-tree $(T,w)$. The main result provides that $w(v)$ is an eigenvalue of $\mathcal{A}(T,w)$ for every $v\in V(T)$, and if $\lambda\neq w(v)$ for every $v\in V(T)$, then $\lambda$ is an eigenvalue of $\mathcal{A}(T,w)$ 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 $\mathcal{A}(T,w)$ 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 (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.