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Summary: On Graphs and Algebraic Graphs that do not Contain
Cycles of Length 4
Noga Alon
H. Tracy Hall
Christian Knauer
Rom Pinchasi§
Raphael Yuster¶
February 5, 2009
Abstract
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 K2,r: rank(A) E-2n(r+1)
r2
n
.
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