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arXiv · 2507.17492

On the sum of the largest and smallest eigenvalues of odd-cycle free graphs

Aida Abiad·Vladislav Taranchuk·Thijs van Veluw
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Abstract

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 $(λ_1+λ_n)/n < 0.0396$.

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Aida Abiad, Vladislav Taranchuk, Thijs van Veluw. 2025-07-23. On the sum of the largest and smallest eigenvalues of odd-cycle free graphs. https://arxiv.org/abs/2507.17492

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