Upper bound on the $k$-th eigenvalue of a graph
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917370270842880 |
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| author | Sivashankar, Varun |
| author_facet | Sivashankar, Varun |
| contents | We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case $k=3$ and generalize the upper bound for all $k \geq 3$ by using the positivity of Gegenbauer polynomials. The upper bound is tight for $k \in \{2,3,4,8,24\}$. We also highlight the close relation of $λ_k(G)$ to questions about equiangular lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Upper bound on the $k$-th eigenvalue of a graph Sivashankar, Varun Combinatorics 05C50, 05C35 We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case $k=3$ and generalize the upper bound for all $k \geq 3$ by using the positivity of Gegenbauer polynomials. The upper bound is tight for $k \in \{2,3,4,8,24\}$. We also highlight the close relation of $λ_k(G)$ to questions about equiangular lines. |
| title | Upper bound on the $k$-th eigenvalue of a graph |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2603.28738 |