A spectral condition for odd cycles in graphs
Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ (G) of its adjacency matrix satisfies μ (G) > sqrt(⌊ n2 / 4 ⌋). Then G contains a cycle of length t for every t ≤ n / 320. This condition is sharp: the complete bipartite graph T2 (n) with parts of size ⌊ n / 2 ⌋ and ⌈ n / 2 ⌉ contains no odd cycles and its largest eigenvalue is equal to sqrt(⌊ n2 / 4 ⌋). This condition is stable: if μ (G) is close to sqrt(⌊ n2 / 4 ⌋) and G fails to contain a cycle of length t for some t ≤ n / 321, then G resembles T2 (n). © 2007 Elsevier Inc. All rights reserved.
Linear Algebra and Its Applications
Nikiforov, V. (2008). A spectral condition for odd cycles in graphs. Linear Algebra and Its Applications, 428 (7), 1492-1498. https://doi.org/10.1016/j.laa.2007.09.029