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arXiv · 2608.03196

Extremal graphs for the $k$-th eigenvalue

Hitesh Kumar·Bojan Mohar·Seyed Ahmad Mojallal·Shivaramakrishna Pragada
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Abstract

For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.

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Hitesh Kumar, Bojan Mohar, Seyed Ahmad Mojallal, Shivaramakrishna Pragada. 2026-08-04. Extremal graphs for the $k$-th eigenvalue. https://arxiv.org/abs/2608.03196

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