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October 8, 2025Open Access

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

Key Points

The research establishes that the ratio of eigenvalues tends to O(k^-1) as the odd girth increases.
For odd girth 7, a strong upper bound is identified, specifically showing (λ_1 + λ_n)/n < 0.0396.
These findings build upon previous work demonstrating limits within triangle-free graphs and their bipartiteness.
The implications suggest significant relationships between graph structure, eigenvalues, and their odd girth characteristics.

Abstract

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.

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