A spectral condition for odd cycles in graphs Original Research Article

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies image. Then G contains a cycle of length t for every tless-than-or-equals, slantn/320 This condition is sharp: the complete bipartite graph T2(n) with parts of size left floorn/2right floor and left ceilingn/2right ceiling contains no odd cycles and its largest eigenvalue is equal to image. This condition is stable: if μ(G) is close to image and G fails to contain a cycle of length t for some tless-than-or-equals, slantn/321, then G resembles T2(n).