The (upper) matching number of graph , sometimes known as the edge independence number, is the size of a maximum independent edge set. Equivalently, it is the degree of the matchinggenerating polynomial
(1)

where is the number of matchings 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 , is the floor function. Equality occurs only for a perfect matching, and graph has a perfect matching iff
(3)

where is the vertex count of .
The matching number of a graph is equal to the independence number of its line graph .
The KönigEgeváry theorem states that the matching number equals the vertex cover number (i.e., size of the smallest minimum vertex cover) are equal for a bipartite graph.
If a graph has no isolated points, then
(4)

where is the matching number, 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"].