A sharp upper bound on the third adjacency 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_ | 1866908905672540160 |
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| author | Tang, Quanyu |
| author_facet | Tang, Quanyu |
| contents | For a graph $G$ of order $n$, let $$ λ_1(G)\ge \cdots \ge λ_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ λ_3(G)\le \frac{n}{3}-1, $$ thereby solving a problem of Nikiforov. The bound is best possible whenever $3\mid n$. Our proof is derived from a more general matrix result: if $A=(a_{ij})$ is a real symmetric matrix of order $n$ with $0\le a_{ij}\le 1$ for all off-diagonal entries and $a_{ii}\ge 0$ for all $i$, then $$ λ_{n-1}(A)+λ_n(A)\ge -\frac{2n}{3}. $$ This in particular confirms a conjecture of Leonida and Li. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A sharp upper bound on the third adjacency eigenvalue of a graph Tang, Quanyu Combinatorics Primary 05C50, Secondary 15A18, 15A42 For a graph $G$ of order $n$, let $$ λ_1(G)\ge \cdots \ge λ_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ λ_3(G)\le \frac{n}{3}-1, $$ thereby solving a problem of Nikiforov. The bound is best possible whenever $3\mid n$. Our proof is derived from a more general matrix result: if $A=(a_{ij})$ is a real symmetric matrix of order $n$ with $0\le a_{ij}\le 1$ for all off-diagonal entries and $a_{ii}\ge 0$ for all $i$, then $$ λ_{n-1}(A)+λ_n(A)\ge -\frac{2n}{3}. $$ This in particular confirms a conjecture of Leonida and Li. |
| title | A sharp upper bound on the third adjacency eigenvalue of a graph |
| topic | Combinatorics Primary 05C50, Secondary 15A18, 15A42 |
| url | https://arxiv.org/abs/2603.21181 |