# 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 in (McNulty & Shallon 1983), 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

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

## 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 (Davey et al. 2000), equational theories (Pöschel 1989), flatness (Delić 2001), groupoid rings (Lee 1991), topologies (Lee 1988), varieties (Oates-Williams 1984), finite state automata (Kelarev, Miller & Sokratova 2005), finite state machines (Kelarev & Sokratova 2003), tree languages and tree automata (Kelarev & Sokratova 2001) etc.

## References

• Davey, Brian A.; Idziak, Pawel M.; Lampe, William A.; McNulty, George F. (2000), "Dualizability and graph algebras", Discrete Mathematics, 214 (1): 145–172, doi:10.1016/S0012-365X(99)00225-3, ISSN 0012-365X, MR 1743633
• Delić, Dejan (2001), "Finite bases for flat graph algebras", Journal of Algebra, 246 (1): 453–469, doi:10.1006/jabr.2001.8947, ISSN 0021-8693, MR 1872631
• McNulty, George F.; Shallon, Caroline R. (1983), "Inherently nonfinitely based finite algebras", Universal algebra and lattice theory (Puebla, 1982), Lecture Notes in Math., 1004, Berlin, New York: Springer-Verlag, pp. 206–231, doi:10.1007/BFb0063439, MR 0716184
• Kelarev, A.V. (2003), Graph Algebras and Automata, New York: Marcel Dekker, ISBN 0-8247-4708-9, MR 2064147
• Kelarev, A.V.; Sokratova, O.V. (2003), "On congruences of automata defined by directed graphs", Theoretical Computer Science, 301 (1&ndash, 3): 31&ndash, 43, doi:10.1016/S0304-3975(02)00544-3, ISSN 0304-3975, MR 1975219
• Kelarev, A.V.; Miller, M.; Sokratova, O.V. (2005), "Languages recognized by two-sided automata of graphs", Proc. Estonian Akademy of Science, 54 (1): 46&ndash, 54, ISSN 1736-6046, MR 2126358
• Kelarev, A.V.; Sokratova, O.V. (2001), "Directed graphs and syntactic algebras of tree languages", J. Automata, Languages & Combinatorics, 6 (3): 305&ndash, 311, ISSN 1430-189X, MR 1879773
• Kiss, E.W.; Pöschel, R.; Pröhle, P. (1990), "Subvarieties of varieties generated by graph algebras", Acta Sci. Math. (Szeged), 54 (1&ndash, 2): 57&ndash, 75, MR 1073419
• Lee, S.-M. (1988), "Graph algebras which admit only discrete topologies", Congr. Numer., 64: 147&ndash, 156, ISSN 1736-6046, MR 0988675
• Lee, S.-M. (1991), "Simple graph algebras and simple rings", Southeast Asian Bull. Math., 15 (2): 117&ndash, 121, ISSN 0129-2021, MR 1145431
• Oates-Williams, Sheila (1984), "On the variety generated by Murskiĭ's algebra", Algebra Universalis, 18 (2): 175–177, doi:10.1007/BF01198526, ISSN 0002-5240, MR 0743465
• Pöschel, R (1989), "The equational logic for graph algebras", Z. Math. Logik Grundlag. Math., 35 (3): 273&ndash, 282, doi:10.1002/malq.19890350311, MR 1000970