In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a branch of mathematics, the rank of an
undirected graph has two unrelated definitions. Let equal the number of
vertices of the graph.
* In the
matrix theory of graphs the rank of an undirected graph is defined as the rank of its
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simple ...
.
:Analogously, the
nullity of the graph is the nullity of its adjacency matrix, which equals .
* In the
matroid theory of graphs the rank of an undirected graph is defined as the number , where is the number of
connected components of the graph. Equivalently, the rank of a graph is the
rank of the oriented
incidence matrix associated with the graph.
[. See in particular the discussion on p. 218.]
:Analogously, the
nullity of the graph is the
nullity of its oriented incidence matrix, given by the formula , 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.
Examples
A sample graph and matrix:
(corresponding to the four edges, e1–e4):
In this example, the matrix theory rank of the matrix is 4, because its column vectors are linearly independent.
See also
*
Circuit rank
In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycle (graph theory), cycles, making ...
*
Cycle rank
*
Nullity (graph theory)
Notes
References
*{{citation, last=Chen, first=Wai-Kai, title=Applied Graph Theory, publisher=North Holland Publishing Company, year=1976, 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.
Algebraic graph theory
Graph connectivity
Graph invariants