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arXiv · 2608.09353

The Sharp Upper Bounds for the Median Eigenvalues of Graphs

Zhengbo Chen·Yuzhenni Wang·Xiao-Dong Zhang
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Abstract

Let $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|\lambda_h|,|\lambda_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-minor-free, then $R(G)\leq\sqrt{5}-1$ with equality attained by an infinite family of outerplanar graphs.Moreover, we show that $R(G)\leq\sqrt{d-2}$ for triangle-free graphs with maximum degree at most $d$ and average degree at most $(d-2)(d^2-2d+2)/(d^2-3d+5)$.

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Zhengbo Chen, Yuzhenni Wang, Xiao-Dong Zhang. 2026-08-10. The Sharp Upper Bounds for the Median Eigenvalues of Graphs. https://arxiv.org/abs/2608.09353

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