Publications de l'Institut Mathématique, Nouvelle Série Vol. 95[109], pp. 189–199 (2014) 

AN ASYMPTOTICALLY TIGHT BOUND ON THE $Q$INDEX OF GRAPHS WITH FORBIDDEN CYCLESVladimir NikiforovDepartment of Mathematical Sciences, University of Memphis, Memphis, USAAbstract: Let $G$ be a graph of order $n$ and let $q(G)$ be the largest eigenvalue of the signless Laplacian of $G$. It is shown that if $k\geq2$, $n>5k^2$, and $q(G)\geq n+2k2$, then $G$ contains a cycle of length $l$ for each $l\in\{3,4,\dots,2k+2\}$. This bound on $q(G)$ is asymptotically tight, as the graph $K_{k}\vee\overline{K}_{nk}$ contains no cycles longer than $2k$ and
Classification (MSC2000): 15A42; 05C50 Full text of the article: (for faster download, first choose a mirror)
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