On the Sum of the Largest and Smallest Eigenvalue of Odd-Cycle Free Fraphs - Abiad, Aida; Taranchuk, Vladislav; van Veluw, Thijs (The Electronic Journal of Combinatorics, 2026)
Let $G$ be a graph with adjacency eigenvalues $λ_1 ≥ … ≥ λ_n$. 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 $(λ_1+λ_n)/n < 0.0396$.