A k-matching in a graph G is a set of k edges, no two of which have a vertex in common (i.e., an independent edge set of size k). Let Φ_k be the number of k-matchings in the graph G, with Φ_0(G) = 1 and Φ_1(G) = m the number of edges of G. Then the matching polynomial is defined by μ(x) = sum_(k = 0)^(ν(G)) (-1)^k Φ_k x^(n - 2k), where n = left bracketing bar G right bracketing bar vertex count of G and ν(G) is the matching number (which satisfies ν(G)<=⌊n/2⌋, where ⌊x⌋ is the floor function). The matching polynomial is also known as the acyclic polynomial (Gutman and Trinajstić 1976, Devillers and Merino 2000), matching defect polynomial, and reference polynomial.
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