This research has been conducted while H. Tracy Hall was visiting the Technion—Israeli Institute of Technology.
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On graphs and algebraic graphs that do not contain cycles of length 4†
Article first published online: 26 OCT 2010
DOI: 10.1002/jgt.20542
© 2010 Wiley Periodicals, Inc.
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How to Cite
Alon, N., Hall, H. T., Knauer, C., Pinchasi, R. and Yuster, R. (2011), On graphs and algebraic graphs that do not contain cycles of length 4. Journal of Graph Theory, 68: 91–102. doi: 10.1002/jgt.20542
- †
Publication History
- Issue published online: 25 AUG 2011
- Article first published online: 26 OCT 2010
- Manuscript Revised: 18 APR 2010
- Manuscript Received: 5 MAY 2007
Funded by
- Israel Science Foundation (to N. A. and R. P.)
- USAIsrael BSF grant (to N. A.)
- Hermann Minkowski Minerva Center for Geometry at Tel Aviv University (to N. A.)
- G.I.F. (to R. P.); German-Israeli Foundation for Scientific Research (to R. P.)
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- Cited By
Keywords:
- algebraic graph;
- minimum rank;
- bipartite;
- square-free;
- square-avoiding;
- graph regularity;
- multiset regularity;
- adjacency rank;
- forbidden subgraph
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 
. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:91-102, 2011