An approximate version of Brouwer's Laplacian conjecture

TL;DR

This work bounds the sums of the largest Laplacian eigenvalues of a graph relative to its edge count, introducing the M_k(G) matrix built from the $k$-th additive compound and a diagonal term. It proves a universal bound $\varepsilon_k(G) \le \max_{|U|=k} |E_G(U)| + (4k-2)\sqrt{k}$, with sharper results for $K_{r+1}$-free and bipartite graphs, and extends the approach to token graphs by showing $\lambda_1(L(F_k(G))) \le |E|+4k-2$ (bipartite: $|E|+2k-1$). A subadditivity framework for $\varepsilon_k$ and its token-graph analogue $\varepsilon_k^T$ underpins the decomposition into stars and matchings, enabling tight-looking bounds in those extremal cases. The results further generalize to the signless Laplacian and suggest approximate resolutions to Brouwer's conjecture in broad graph classes, as well as to related questions for token graphs. Overall, the paper advances spectral techniques that connect additive compound theory, graph decompositions, and extremal graph bounds to obtain near-optimal Laplacian sum bounds.

Abstract

Let $G=(V,E)$ be an $n$-vertex graph, $L(G)\in \mathbb{R}^{n\times n}$ its Laplacian matrix, and let $λ_1(L(G))\ge λ_2(L(G))\ge \cdots\ge λ_n(L(G))=0$ denote its eigenvalues. For $1\le k\le n$, let $\varepsilon_k(G)= \sum_{i=1}^k λ_i(L(G)) -|E|$. 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.

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