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 > 5k2, and q(G) ≥ n + 2k-2, then G contains a cycle of length l for each l ε {3, 4, . . . , 2k + 2}. This bound on q(G) is asymptotically tight, as the graph Kk v K̄n-k contains no cycles longer than 2k and q(Kk v K̄n-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.

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Publications de l'Institut Mathematique