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.
On Brouwer's Laplacian conjecture
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abstract
Brouwer's Laplacian conjecture states that the sum of the largest $k$ eigenvalues of a graph's Laplacian is less than or equal to the number of edges plus $\binom{k+1}{2}$. We give a proof of this conjecture. Our proof relies on the Grone--Merris--Bai theorem for \emph{split} graphs. We also show the converse, thereby establishing an equivalence between Brouwer's conjecture and the Grone--Merris--Bai theorem.
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math.CO 2years
2026 2verdicts
ACCEPT 2representative citing papers
Lew's matching-number and vertex-cover conjectures on sums of the largest Laplacian eigenvalues are proved from Brouwer's inequality.
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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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Proofs of two conjectures on generalizations of Brouwer's Laplacian conjecture
Lew's matching-number and vertex-cover conjectures on sums of the largest Laplacian eigenvalues are proved from Brouwer's inequality.