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

Abstract

The sum $\lambda_1 + \lambda_n$ 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 $k\geq 3$, let $\gamma_k$ denote the supremum of $\frac{\lambda_1 + \lambda_n}{n}$ over graphs without odd cycles of length less than $k$. The example of the $k$-cycle $C_k$ shows that $\gamma_k\geq \Omega(k^{-3})$. In their recent work, Abiad, Taranchuk, and Van Veluw showed that $\gamma_k\leq O(k^{-1})$ and asked to determine the asymptotics of $\gamma_k$. Using approximation theory, we show that $\gamma_k\leq O(k^{-3}(\log k)^3)$, giving a tight upper bound up to a poly-logarithmic factor.