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

Let $G$ be a graph with adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Both $\lambda_1 + \lambda_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikv\'ari proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that $(\lambda_1+\lambda_n)/n \le (3-2\sqrt 2) < 0.1716$. In this paper we extend Csikv\'ari's result to general odd girth $k$ proving that $(\lambda_1+\lambda_n)/n = O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $(\lambda_1+\lambda_n)/n < 0.0396$.