The spectral radius of graphs without long cycles.

Abstract Let C ℓ be a cycle of length ℓ and S n , k = K k ∨ K n − k ‾ , the join graph of a complete graph of order k and an empty graph on n − k vertices, and S n , k + be the graph obtained from S n , k by adding an edge in the independent set of S n , k. Nikiforov conjectured that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains C 2 k + 1 or C 2 k + 2 (or C 2 k + 2), unless G = S n , k (or G = S n , k +). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains a cycle C ℓ with ℓ ≥ 2 k + 1 (or C ℓ with ℓ ≥ 2 k + 2), unless G = S n , k (or G = S n , k +). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2]. [ABSTRACT FROM AUTHOR]