A spectral condition for odd cycles in graphs

Abstract Let G be a graph of sufficiently large order n , and let the largest eigenvalue μ ( G ) of its adjacency matrix satisfies μ ( G ) > ⌊ n 2 / 4 ⌋ . Then G contains a cycle of length t for every t ⩽ n / 320 This condition is sharp: the complete bipartite graph T 2 ( n ) with parts of size ⌊ n / 2 ⌋ and ⌈ n / 2 ⌉ contains no odd cycles and its largest eigenvalue is equal to ⌊ n 2 / 4 ⌋ . This condition is stable: if μ ( G ) is close to ⌊ n 2 / 4 ⌋ and G fails to contain a cycle of length t for some t ⩽ n / 321 , then G resembles T 2 ( n ) .

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