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

Noga Alon, H. Tracy Hall, Christian Knauer, Rom Pinchasi, Raphael Yuster
2010 Journal of Graph Theory  
We consider extremal problems for algebraic graphs, that is, graphs whose vertices correspond to vectors in R d , 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
more » ... h does not contain a copy of K 2,r : rank(A) ≥ E−2n(r+1) r 2 √ n .
doi:10.1002/jgt.20542 fatcat:fkjquevm3bag7n4f5b6wzf4dum