An overview of the Matching Polynomial

A $k$-matching in a graph $X$ is a set of $k$ vertex-disjoint edges. If we denote

the number of $k$-matchings in a graph $G$ by $p(G,k)$ and $|V(G)|=n$, then its matching polynomial is

\[

\mu(G,t) = \sum_k (-1)^k p(G,k) t^{n-2k}.

\]

It is thus a form of generating function, with fudge factors inserted to make our work easier.

Matchings are a central topic in graph theory, but nonetheless this polynomial appeared in Physics and in Chemistry, before becoming an object of interest to graph theorists. If the graph $G$ is a forest, its matching polynomial coincides with the characteristic polynomial of the adjacency matrix of $G$, and considering the analogies between these two polynomials has proved very fruitful. In my talk I will present some of the history of the matching polynomial, along with interesting parts of its theory.

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