An Asymptotically Tight Bound on the Q-index of Graphs with Forbidden Cycles

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 > 2, n > 5k, and q(G) > n + 2k − 2, then G contains a cycle of length l for each l ∈ {3, 4, . . . , 2k + 2}. This bound on q(G) is asymptotically tight, as the graph Kk ∨Kn−k contains no cycles longer than 2k and q(Kk ∨ Kn−k) > n + 2k − 2− 2k(k − 1) n + 2k − 3 . The main result gives an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with forbidden cycles. The proof of the main result and the tools used therein could serve as a guidance to the proof of the full conjecture.