The (upper) matching number 
of a graph 
, sometimes known as the edge independence number, is the size
of a maximum independent edge set.
Equivalently, it is the degree of the matching-generating
polynomial
 |
(1)
|
where 
is the number of matchings of size 
of a graph 
. The notations 
, 
, or 
are sometimes also used.
The matching number is also the size of a largest maximal independent edge set, while the size of a smallest maximal independent edge set is called the lower matching number.
satisfies
 |
(2)
|
where 
is the vertex count of 
and 
is the floor function.
A graph 
has a perfect matching iff
 |
(3)
|
where 
is the vertex count of 
.
If a graph 
on 
vertices is traceable,
then every other edge of a Hamiltonian
path gives a matching of size 
. Thus traceability
implies 
.
The matching number 
of a graph 
is equal to the independence
number 
of its line graph 
.
The König-Egerváry theorem states that the matching number equals the vertex cover number (i.e., size of a minimum vertex cover) for a bipartite graph.
If a graph 
has no isolated vertices,
then
 |
(4)
|
where 
is the size of a minimum edge cover and 
is the vertex
count of 
(West 2000).
Precomputed matching numbers for many named graphs are available in the Wolfram Language using GraphData[graph, "MatchingNumber"].
