Graph algebra

In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon,Template:Sfn and has seen many uses in the field of universal algebra since then.

Definition

Let D = (V, E) be a directed graph, and 0 an element not in Template:Mvar. The graph algebra associated with Template:Mvar has underlying set $V\cup \{0\}$, and is equipped with a multiplication defined by the rules

• xy = x if $x,y\in V$ and $(x,y)\in E$,
• xy = 0 if $x,y\in V\cup \{0\}$ and $(x,y)\notin E$.

Applications

This notion has made it possible to use the methods of graph theory in universal algebra and several other areas of discrete mathematics and computer science. Graph algebras have been used, for example, in constructions concerning dualities,Template:Sfn equational theories,Template:Sfn flatness,Template:Sfn groupoid rings,Template:Sfn topologies,Template:Sfn varieties,Template:Sfn finite-state machines,Template:SfnTemplate:Sfn tree languages and tree automata,Template:Sfn etc.

See also

• Group algebra (disambiguation)
• Incidence algebra
• Path algebra