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

Let $G$ be a graph with adjacency eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$. Both ${\lambda }_1+{\lambda }_n$ 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 $({\lambda }_1+{\lambda }_n)\mathrm{/}n\le (3-2\sqrt{2})<0.1716$. In this paper we extend Csikvári’s result to general odd girth $k$ proving that $({\lambda }_1+{\lambda }_n)\mathrm{/}n=O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $({\lambda }_1+{\lambda }_n)\mathrm{/}n<0.0396$.