Eigenvalue monotonicity of $q$-Laplacians of trees along a poset

TL;DR

This paper investigates how the eigenvalues of the $q$-Laplacian matrix of trees change along a specific poset, revealing monotonic behavior of spectral properties as trees are ordered.

Contribution

It generalizes known eigenvalue monotonicity results for Laplacians to $q$-Laplacians of trees within the generalized tree shift poset.

Findings

Spectral radius and second smallest eigenvalue increase along the poset

Smallest eigenvalue decreases along the poset

Results extend to $q,t$-Laplacians and exponential distance matrices

Abstract

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.