Spectra of weighted uniform hypertrees

TL;DR

This paper studies the eigenvalues of the adjacency tensor of weighted uniform hypertrees, revealing that vertex weights are eigenvalues and linking other eigenvalues to roots of a weighted matching polynomial, extending prior spectral results.

Contribution

It extends spectral analysis of hypertrees to weighted cases, characterizing eigenvalues via a weighted matching polynomial and connecting spectral radius to polynomial roots.

Findings

Vertex weights are eigenvalues of the adjacency tensor.

Eigenvalues not equal to vertex weights correspond to roots of the matching polynomial of subtrees.

Spectral radius equals the largest root of the matching polynomial for real, nonnegative weights.

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.