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 
.
