A spectral condition for odd cycles in graphs

Let G be a graph of su¢ ciently large order n; and let the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c: 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 bn=2c and dn=2e contains no odd cycles and its largest eigenvalue is equal to p bn2=4c: This condition is stable: if (G) is close to p bn2=4c and G fails to contain a cycle of length t for some t n=321; then G resembles T2 (n) : Keywords: odd cycle; triangle; graph spectral radius; stabilty AMS classi…cation: 05C50, 05C35.. Introduction This note is part of an ongoing project aiming to build extremal graph theory on spectral grounds, see, e.g., [3] and [6, 13]. It is known ([9], [14]) that if G is a graph of order n and the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c; then a triangle exists in G. Here we show that the same premises imply the existence of other cycles as well. Theorem 1 Let G be a graph of su¢ ciently large order n with (G) > p bn2=4c: Then G contains a cycle of length t for every t n=320: Write T2 (n) for the complete bipartite graph with parts of size bn=2c and dn=2e : Note that T2 (n) contains no odd cycles and (T2 (n)) = p bn2=4c; thus, Theorem 1 gives a sharp spectral condition for the existence of odd cycles. Moreover, there is stability in this condition: if (G) is close to p bn2=4c and G fails to contain a cycle of length t for some t n=321; then G resembles T2 (n) : Here is a precise form of this statement.