Maxima of the Q-index: Forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q (G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let Fn be the friendship graph of order n; if n is even, let Fn be Fn-1 with an extra edge hung to its center. It is shown that if G is a graph of order n ≥ 4, with no 4-cycle, then q (G) \u3c q (Fn), unless G = Fn. Let Sn,k be the join of a complete graph of order k and an independent set of order n - k. It is shown that if G is a graph of order n ≥ 6, with no 5-cycle, then q (G) \u3c q (Sn,2), unless G = Sn,k. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q (G) of graphs with forbidden cycles