  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.
