An approximate version of Brouwer's Laplacian conjecture

TL;DR

This paper provides approximate bounds on the eigenvalues of a graph's Laplacian matrix, improving known bounds and addressing conjectures related to Brouwer's Laplacian conjecture and token graphs.

Contribution

It introduces new bounds on Laplacian eigenvalues that approximate Brouwer's conjecture and extends to signless Laplacians and token graphs.

Findings

Bound on b5_k(G) q max_{UV, |U|=k} |E_G(U)| + (4k-2)sqrt{k}

Improved bounds for large k,  inom{k}{2} + (4k-2)sqrt{k}

Bound on the largest Laplacian eigenvalue of token graphs by |E| + 4k - 2

Abstract

Let $G=(V,E)$ be an $n$-vertex graph, $L(G)\in \mathbb{R}^{n\times n}$ its Laplacian matrix, and let $\lambda_1(L(G))\ge \lambda_2(L(G))\ge \cdots\ge \lambda_n(L(G))=0$ denote its eigenvalues. For $1\le k\le n$, let $\varepsilon_k(G)= \sum_{i=1}^k \lambda_i(L(G)) -|E|$. We show that for every $1\le k\le n$, \[ \varepsilon_k(G) \le \max_{U\subset V,\, |U|=k} |E_G(U)| + (4k-2)\sqrt{k}, \] where $E_G(U)$ is the set of edges of $G$ contained in $U$. As an immediate consequence, we obtain that $\varepsilon_k(G)\le \binom{k}{2}+(4k-2)\sqrt{k}$. This improves upon previously known bounds for large values of $k$, and may be seen as an approximate version of a conjecture of Brouwer, stating that $\varepsilon_k(G)\le \binom{k+1}{2}$ for every graph $G$. Moreover, for every $r\ge 2$, if $G$ is a $K_{r+1}$-free graph, we obtain that $\varepsilon_k(G)\le (1-1/r)k^2/2 + (4k-2)\sqrt{k}$, which is tight up to the sub-quadratic term. Our arguments rely on the study of the largest eigenvalue of a matrix obtained by performing a certain diagonal perturbation on the $k$-th additive compound matrix of $L(G)$. Using similar methods, we show that the largest Laplacian eigenvalue of the $k$-th token graph of a graph $G=(V,E)$ is bounded from above by $|E|+4k-2$, obtaining a weak version of a conjecture of Apte, Parekh, and Sud, which predicts that an upper bound of $|E|+k$ should hold. All our results also hold, with essentially the same proofs, when the Laplacian matrix is replaced by the signless Laplacian of the graph.