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

Abstract

The sum $λ_1 + λ_n$ 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 $γ_k$ denote the supremum of $\frac{λ_1 + λ_n}{n}$ over graphs without odd cycles of length less than $k$. The example of the $k$-cycle $C_k$ shows that $γ_k\geq Ω(k^{-3})$. In their recent work, Abiad, Taranchuk, and Van Veluw showed that $γ_k\leq O(k^{-1})$ and asked to determine the asymptotics of $γ_k$. Using approximation theory, we show that $γ_k\leq O(k^{-3}(\log k)^3)$, giving a tight upper bound up to a poly-logarithmic factor.

Theorems & Definitions (25)

• Theorem 1.2: Abiad, Taranchuk, and Van Veluw abiad2025, Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Proposition \ref{['broad spectrum']}
• proof : Proof of Proposition \ref{['high lambda_1']}
• proof : Proof of Theorem \ref{['main']}