Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling
Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.