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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.07494 |
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| _version_ | 1866916564927774720 |
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| author | Li, Sida |
| author_facet | Li, Sida |
| contents | Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $λ_1 \ge λ_2 \ge \dots \ge λ_n$. The problem of bounding $λ_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov, who demonstrated strong upper and lower bounds for arbitrary $k$. Nikiforov also claimed a strengthened upper bound for $k \ge 3$, namely that $\frac{λ_k}{n} < \frac{1}{2\sqrt{k-1}} - \varepsilon_k$ for some positive $\varepsilon_k$, but omitted the proof due to its length. In this paper, we give a proof of this bound for $k = 3$. We achieve this by instead looking at $λ_{n-1} + λ_n$ and introducing a new graph operation which provides structure to minimising graphs, including $ω\le 3$ and $χ\le 4$. Then we reduce the hypothetical worst case to a graph that is $n/2$-regular and invariant under said operation. By considering a series of inequalities on the restricted eigenvector components, we prove that a sequence of graphs with $\frac{λ_{n-1} + λ_n}{n}$ converging to $-\frac{\sqrt{2}}{2}$ cannot exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strengthened upper bound on the third eigenvalue of graphs Li, Sida Combinatorics Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $λ_1 \ge λ_2 \ge \dots \ge λ_n$. The problem of bounding $λ_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov, who demonstrated strong upper and lower bounds for arbitrary $k$. Nikiforov also claimed a strengthened upper bound for $k \ge 3$, namely that $\frac{λ_k}{n} < \frac{1}{2\sqrt{k-1}} - \varepsilon_k$ for some positive $\varepsilon_k$, but omitted the proof due to its length. In this paper, we give a proof of this bound for $k = 3$. We achieve this by instead looking at $λ_{n-1} + λ_n$ and introducing a new graph operation which provides structure to minimising graphs, including $ω\le 3$ and $χ\le 4$. Then we reduce the hypothetical worst case to a graph that is $n/2$-regular and invariant under said operation. By considering a series of inequalities on the restricted eigenvector components, we prove that a sequence of graphs with $\frac{λ_{n-1} + λ_n}{n}$ converging to $-\frac{\sqrt{2}}{2}$ cannot exist. |
| title | Strengthened upper bound on the third eigenvalue of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.07494 |