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Maximum spectral sum of graphs

Hitesh Kumar, Lele Liu, Hermie Monterde, Shivaramakrishna Pragada, Michael Tait · 2026 · math.CO · DOI 10.48550/arxiv.2604.00512 · arXiv 2604.00512

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abstract

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum \[ \lambda_1(G) + \lambda_2(G)\le \frac{8}{7}n \] for any graph $G$. We prove this conjecture by combining tools from the theory of graph limits, convex geometry, exterior algebra and convex optimization. The techniques developed are of independent interest.

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2026 1

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Graph Eigenvalues and Projection Constants

math.CO · 2026-08-03 · conditional · novelty 7.0

λ_k(G) ≤ ((k−2)√(k+1)+2)n/(2k(k−1)) − 1 for all graphs, tight for k ∈ {2,3,4,8,24}, resolving c₃ = 1/3 and Nikiforov's Conjecture 4.2.

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  • Graph Eigenvalues and Projection Constants math.CO · 2026-08-03 · conditional · none · ref 11 · internal anchor

    λ_k(G) ≤ ((k−2)√(k+1)+2)n/(2k(k−1)) − 1 for all graphs, tight for k ∈ {2,3,4,8,24}, resolving c₃ = 1/3 and Nikiforov's Conjecture 4.2.