Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is claw free with no induced 4-cycle in its complement. For the last family we show that the second power of the edge ideal has a linear resolution.