Rank (graph theory)

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In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph.

Analogously, the nullity of the graph is the nullity of its adjacency matrix, which equals nr.
Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula mn + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti number of the graph. The sum of the rank and the nullity is the number of edges.

See also[edit]


  1. ^ Weisstein, Eric W. "Graph Rank." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GraphRank.html
  2. ^ Grossman, Jerrold W.; Kulkarni, Devadatta M.; Schochetman, Irwin E. (1995), "On the minors of an incidence matrix and its Smith normal form", Linear Algebra and its Applications, 218: 213–224, doi:10.1016/0024-3795(93)00173-W, MR 1324059 . See in particular the discussion on p. 218.


  • Chen, Wai-Kai (1976), Applied Graph Theory, North Holland Publishing Company, ISBN 0-7204-2371-6 .
  • Hedetniemi, S. T., Jacobs, D. P., Laskar, R. (1989), Inequalities involving the rank of a graph. Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 6, pp. 173–176.
  • Bevis, Jean H., Blount, Kevin K., Davis, George J., Domke, Gayla S., Miller, Valerie A. (1997), The rank of a graph after vertex addition. Linear Algebra and its Applications, vol. 265, pp. 55–69.