Given a directed graph with a finite number of vertices and edges, a path algebra can be produced from the linear combinations of vertices and paths. In order to better understand path algebras, we wanted to create an isomorphism between arbitrary path algebra and a ring of matrices. We started by looking at the simple case of two vertices linked by one directed path, and found it to be isomorphic to the ring of 2×2 lower triangular matrices. This result motivated our primary goal of finding an isomorphism for path algebras associated with directed graphs with more vertices and edges. In particular we discovered an explicit isomorphism for directed graphs without cycles, including graphs with multiple paths between particular pairs of vertices.

**Lucio Tolentino, ’09 **Corona, CA

Majors: Computer Science, Mathematics

**Benjamin Greenberg**

Grinnell College

**David MacFadden**

Augustana College

Sponsor: James Freeman