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 (3-22) < 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.