# Graph algebra

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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 in Template:Harv, and has seen many uses in the field of universal algebra since then.

## Definition

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

## Applications

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

## References

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