Proof of Brouwers Conjecture (BC) for all graphs with number of vertices n > n_0 assuming that BC holds for n< n_0 for some n_0

TL;DR

The paper investigates Brouwer's Conjecture on the sum of the $t$ largest Laplacian eigenvalues, proving it for all graphs with $n>n_0$ assuming validity for all graphs with $n\le n_0$ (with $n_0$ up to $10^{24}$). The approach combines induction on the number of vertices, a Grassmannian-based orthonormal basis adapted to the Laplacian eigenbasis, and duality with the graph complement, to bound $S_t(G)$ via $S_{t-1}(G-\{1\})$ and perturbative terms from distinguished vectors $x_1$ and $x_{t+1}$. It leverages the GMB inequality and the threshold-graph framework, showing that spectrally threshold dominated graphs inherit Conjecture validity through a construction that attains Brouwer's bound on threshold graphs. By establishing propagation steps through cases on $x_{1,1}^2$ and edge-structure quantities, the authors reduce the problem to threshold graphs and their thresholds in $\mathcal{T}(G)$, culminating in a conditional universal BC proof. The work illuminates the connection between spectral bounds and extremal threshold-graph structures, and indicates that tightening the induction bounds could remove the conditional nature in future work.

Abstract

Assuming that Brouwers Conjecture the upper bound for the sum of t< n largest eigenvalues of Laplacian graph on n vertices true for n <n_0, we prove the Brouwers Conjecture BC for n > n_0 for some fixed n_0

Theorems & Definitions (13)

• Conjecture 1
• theorem 1
• theorem 2: 3
• lemma 1
• lemma 2
• proposition 1
• proof
• lemma 3
• proof
• proposition 2