Remarks on the Brouwer Conjecture

2025/08/11 by Knill, Oliver
#05Cxx 05Exx 68Rxx #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.07550

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.

Citations

• Sums of Laplacian eigenvalues and sums of degrees
• Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs
• Eigenvalue bounds of the Kirchhoff Laplacian
• Constraints on Brouwer's Laplacian Spectrum Conjecture
• Brouwer's conjecture holds asymptotically almost surely
• The Laplacian Spectrum of a Graph II

Related

• Proof of Brouwers Conjecture (BC) for all graphs with number of vertices n > n₀ assuming that BC holds for n< n₀ for some n₀
• More on the full Brouwer Laplacian spectrum conjecture
• Constraints on Brouwer's Laplacian Spectrum Conjecture
• Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues
• A partial proof of the Brouwer's conjecture