On graphs and algebraic graphs that do not contain cycles of length 4

We consider extremal problems for algebraic graphs, that is, graphs whose vertices correspond to vectors in ℝd, where two vectors are connected by an edge according to an algebraic condition. We also derive a lower bound on the rank of the adjacency matrix of a general abstract graph using the number of 4-cycles and a parameter which measures how close the graph is to being regular. From this, we derive a rank bound for the adjacency matrix A of any simple graph with n vertices and E edges which does not contain a copy of **image**. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:91-102, 2011 (This research has been conducted while H. Tracy Hall was visiting the Technion—Israeli Institute of Technology.)