A partial proof of the Brouwer's conjecture

2024/12/17 by Filipovski, Slobodan
#05C50 #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2412.12952

Abstract

Let G be a simple graph with n vertices and m edges and let k be a natural number such that k≤ n. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most m+k+1 \choose 2. In this paper we prove that this conjecture is true for simple (m,n)-graphs where n≤ m≤ (√(3)-1)/(4)(n-1)n and k∈ [ √[3]\frac8m²n-1+4mn+n², n]. Moreover, we prove that the conjecture is true for all simple (m,n)-graphs where k (≤ n) is a natural number from the interval [√2n-2m+2√2m²+mn(n-1),1+\frac8m²n²(n-1)+(4m)/(n)].

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
• Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues
• Constraints on Brouwer's Laplacian Spectrum Conjecture
• More on the full Brouwer Laplacian spectrum conjecture
• Remarks on the Brouwer Conjecture