Somewhat related to this, you have the rather new field of Leavitt Path Algebras, which takes a field $K$ and a directed graph $E$, generates its extended graph $E'$ (add to $E$ its own edges reversed), and finally computes the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called (CK1) and (CK2), inherited from the $C*$-algebras setting.

These associative algebras provide us simultaneously with a purely algebraic analog of $C*$-algebras of graph and a generalization of the Leavitt algebras (associative algebras which do not satisfy the IBN property).

The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.

Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.

The theory of LPAs is a beautiful one because it allows us to identify ring-theoretic properties of associative algebras from the graph-theoretic properties of their associated graphs in a visual and straightforward way.

Some references:

G. Abrams, G. Aranda Pino. "The Leavitt path algebra of a graph", J. Algebra 293 (2), 319-334 (2005). (Available at http://agt.cie.uma.es/~gonzalo/papers/AA1_Web.pdf).

P. Ara, M.A. Moreno, E. Pardo. "Nonstable K-Theory for graph algebras", Algebra Repr. Th. DOI 10.1007/s10468-006-9044-z (electronic).
(Available at http://www.springerlink.com/content/pu701474q5300m63/).

G. Abrams, G. Aranda Pino, F. Perera, M. Siles Molina. "Chain conditions for Leavitt path algebras".
(Available at http://agt.cie.uma.es/~gonzalo/papers/AAPS1_Web.pdf).

K.R. Goodearl. "Leavitt path algebras and direct limits", Contemp. Math. 480 (2009), 165-187.

maths, no need to be overly humble. – Scott Morrison♦ Oct 27 '09 at 1:33