Remarks on the Brouwer Conjecture

TL;DR

This paper investigates the Brouwer conjecture in spectral graph theory, establishing conditions under which it holds for large graphs and extending its implications to quivers, thus advancing understanding of eigenvalue bounds.

Contribution

The paper proves the Brouwer conjecture for graphs with sufficiently large vertex count relative to maximum degree and links its validity to quivers.

Findings

BC holds for graphs with n ≥ 4 * (max degree)^2 vertices.

BC for graphs implies BC for quivers.

Provides bounds relating eigenvalues and graph parameters.

Abstract

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n is larger or equal than 4 times the square of the maximal vertex degree. We also note that (BC) for graphs implies (BC) for quivers.