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
F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1}
with an edge hanged to its center. It is shown that if G is a graph of order n,
with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,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, with no 5-cycle, then q(G)<q(S_{n,2}), unless
G=S_{n,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.Comment: 12 page
