The word "rank" refers to several related concepts in mathematics involving elliptic curves, 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, ... is 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 
.
