On the sum of the largest and smallest eigenvalues of graphs with high odd girth

The sum λ₁ + λₙ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer k 3, let γₖ denote the supremum of λ₁ + λₙn over graphs without odd cycles of length less than k. The example of the k-cycle Cₖ shows that γₖ Ω (k^-3). In their recent work, Abiad, Taranchuk, and van Veluw showed that γₖ O (k^-1) and asked to determine the asymptotics of γₖ. Using approximation theory, we show that γₖ O (k^-3³ k), giving a tight upper bound up to a poly-logarithmic factor.