Perfect matchings and cluster algebras of classical type

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases.