Matching polynomial

The matching polynomial of a graph encodes the number of its matchings of each size as coefficients, commonly with alternating signs by matching size.

Matching (graph theory): A set of edges in a graph no two of which share a vertex. Each coefficient counts matchings of a particular size.
Matching number: The largest size of a matching in a graph. It determines the degree of the matching polynomial in the alternating-sign convention.
Independence polynomial: A graph polynomial whose coefficients count independent vertex sets by size. It counts vertex sets rather than edge matchings.
Monomer–dimer model: A statistical-mechanical model of particles occupying vertices singly or covering adjacent vertex pairs in a graph or lattice. Its partition function is the matching generating polynomial with a dimer activity.
Henri Heilmann: A mathematician who, with Elliott Lieb, proved fundamental results on zeros of matching polynomials. His work established the theorem governing the polynomial's real zeros.
Induced subgraph: A graph formed from a vertex subset by retaining every edge joining two selected vertices. Deleting vertices produces the smaller graphs used in recurrence relations.
Deletion-contraction recurrence: A recurrence that relates a graph invariant to values on graphs obtained by deleting an edge or its endpoints. Deleting a vertex separates matchings according to whether they cover it.
Matching generating function: A polynomial or generating series whose coefficients count matchings by size, often without alternating signs. It encodes the same counts with a convention that differs from the signed matching polynomial.
Dimer model: A model in which dimers occupy pairs of adjacent sites, often studied through perfect matchings. Perfect matchings are the top-size terms in the matching data.
Elliott Lieb: A mathematical physicist known for major contributions to quantum mechanics and statistical mechanics. His collaboration with Heilmann connected matching polynomials to dimer models.

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