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

Let G be a graph with adjacency eigenvalues λ₁ ≥ ⋯ ≥ λₙ. Both λ₁ + λₙ 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 (λ₁+λₙ)/n ≤ (3-2√ 2) < 0.1716. In this paper we extend Csikvári's result to general odd girth k proving that (λ₁+λₙ)/n = O(k⁻¹). In the case of odd girth 7, we prove a stronger upper bound of (λ₁+λₙ)/n < 0.0396.