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.