The word "rank" refers to several related concepts in mathematics involving graphs, groups, matrices, quadratic forms, sequences, set theory, statistics, and tensors.
In graph theory, the graph rank of a graph is defined as , where is the number of vertices on and is the number of connected components (Biggs 1993, p. 25).
In set theory, rank is a (class) function from sets to ordinal numbers. The rank of a set is the least ordinal number greater than the rank of any member of the set (Mirimanoff 1917; Moore 1982, pp. 261-262; Rubin 1967, p. 214). The proof that rank is well-defined uses the axiom of foundation.
For example, the empty set has rank 0 (since it has no members and 0 is the least ordinal number), has rank 1 (since , its only member, has rank 0), has rank 2, and has rank . Every ordinal number has itself as its rank.
Mirimanoff (1917) showed that, assuming the class of urelements is a set, for any ordinal number , the class of all sets having rank is a set, i.e., not a proper class (Rubin 1967, p. 216) The number of sets having rank for , 1, ... are 1, 1, 2, 12, 65520, ... (OEIS A038081), and the number of sets having rank at most is , 1, 2, 4, 16, 65536, ... (OEIS A014221).
The rank of a mathematical object is defined whenever that object is free. In general, the rank of a free object is the cardinal number of the free generating subset .