On the sum of the largest and smallest eigenvalues of odd-cycle free graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909702282018816 |
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| author | Abiad, Aida Taranchuk, Vladislav van Veluw, Thijs |
| author_facet | Abiad, Aida Taranchuk, Vladislav van Veluw, Thijs |
| contents | Let $G$ be a graph with adjacency eigenvalues $λ_1 \geq \cdots \geq λ_n$. Both $λ_1 + λ_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikvári proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that $(λ_1+λ_n)/n \le (3-2\sqrt 2) < 0.1716$. In this paper we extend Csikvári's result to general odd girth $k$ proving that $(λ_1+λ_n)/n = O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $(λ_1+λ_n)/n < 0.0396$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sum of the largest and smallest eigenvalues of odd-cycle free graphs Abiad, Aida Taranchuk, Vladislav van Veluw, Thijs Combinatorics Let $G$ be a graph with adjacency eigenvalues $λ_1 \geq \cdots \geq λ_n$. Both $λ_1 + λ_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikvári proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that $(λ_1+λ_n)/n \le (3-2\sqrt 2) < 0.1716$. In this paper we extend Csikvári's result to general odd girth $k$ proving that $(λ_1+λ_n)/n = O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $(λ_1+λ_n)/n < 0.0396$. |
| title | On the sum of the largest and smallest eigenvalues of odd-cycle free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.17492 |