An asymptotically tight bound on the Q-index of graphs with forbidden cycles

V. Nikiforov

Published 2013-10-05, updated 2014-02-25Version 2

Let G be a graph of order n and let q(G) be that largest eigenvalue of the signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2, then G contains cycles of length l whenever 2<l<2k+3. This bound is asymptotically tight. It implies an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with no cycle of a specified length.