The inequality ε_k(G) ≤ kν(G) holds for all graphs in the range 1≤k≤n(G)−2, resolving Lew's conjecture, with all equality cases characterized.
More on the full Brouwer Laplacian spectrum conjecture
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abstract
Brouwer conjectured that the sum of the first $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for each $k\in \{1,2,\cdots,n\}$, which has come to be known as Brouwer's conjecture. Recently, Li and Guo further considered the case when the equalities hold in these conjectured inequalities, and proposed the full version of Brouwer's conjecture. In this paper, we first present a concise version of the full Brouwer's conjecture. Then we show that the full Brouwer's conjecture holds for two families of spanning subgraphs of complete split graphs and for $c$-cyclic graphs with $c\in\{0,1,2\}$. We also consider the Nordhaus-Gaddum version of the full Brouwer's conjecture and present partial solutions to it.
fields
math.CO 2years
2026 2representative citing papers
New upper bounds for sum of k largest eigenvalues of symmetric matrices improve Mohar's graph bound and partially resolve Brouwer's Laplacian conjecture for small k on almost all graphs.
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A Matching-Number Refinement of Brouwer's Laplacian Eigenvalue Inequality
The inequality ε_k(G) ≤ kν(G) holds for all graphs in the range 1≤k≤n(G)−2, resolving Lew's conjecture, with all equality cases characterized.
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Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications
New upper bounds for sum of k largest eigenvalues of symmetric matrices improve Mohar's graph bound and partially resolve Brouwer's Laplacian conjecture for small k on almost all graphs.