The concept of a graph algebra was introduced by G.F. McNulty and C.R. Shallon in [2]. Let ${\displaystyle D=(V,E)}$ be a directed graph (see Graph (data structure)), and let ${\displaystyle 0}$ be 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 ${\displaystyle xy=x}$ if ${\displaystyle x,y\in V,(x,y)\in E}$, and ${\displaystyle xy=0}$ if ${\displaystyle x,y\in V\cup \{0\},(x,y)\notin E}$.

Graph algebras have been used in several directions of mathematical research as a convenient source of examples required for the proofs of various theorems (see, for example, [1-5]).

References

[1] B.A. Davey, P.M. Idziak, W.A. Lampe and G. F. McNulty, Dualizability and graph algebras, Discrete Math. 214 (1-3) (2000), 145-172.

[2] G.F. McNulty and C.R. Shallon, Inherently nonfinitely based finite algebras, Universal Algebra and Lattice Theory (Puebla, 1982), Springer, Berlin, 1983, 206-231.

[3] A.V. Kelarev, "Graph Algebras and Automata", Marcel Dekker, New York, 2003. ISBN: 0-8247-4708-9.

[4] A.V. Kelarev and O.V. Sokratova, On congruences of automata defined by directed graphs, Theoretical Computer Science 301 (2003), 31-43.

[5] E.W. Kiss, R. P"oschel, and P. Pr"ohle, Subvarieties of varieties generated by graph algebras, Acta Sci. Math. (Szeged) 54(1-2) (1990), 57-75.