Year: 2026
Type: preprint
Abstract: Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^+(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^+(G)\le m+\binom{k+1}{2}$ for $k=1, \ldots, n$. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if $k=1$ and $G$ is either star $K_{1,a}$ or triangle $K_3$ with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.
Source: arXiv (Cornell University)
Authors: Yuan, Ruisong, Zhang, Xiao-Dong
Institutions: Shanghai Jiao Tong University, Shanghai Innovation Institute
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1413495475, G3864957230
OpenAlex ID: W7214610708