The finite graph theory in a directed graph form a mathematics category under concatenation called the free object category (which is generated by a graph). The length of a path in gives a functor from this category into the natural numbers . A k-graph is a natural generalistion of this concept which was introduced in by Alex Kumjian and David Pask.
In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category with domain and codomain maps and , together with a functor which satisfies the following factorisation property: if then there are unique with such that .
- It can be shown that a 1-graph is precisely the path category of a directed graph.
- The category consisting of a single object and k commuting morphisms , together with the map defined , is a k-graph.
- Let then is a k-graph when gifted with the structure maps , , and .
The notation for k-graphs is borrowed extensively from the corresponding notation for categories:
- For let .
- By the factorisation property it follows that .
- For and we have , and .
- If for all and then is said to be row-finite with no sources.
Visualisation - Skeletons
A k-graph is best visualised by drawing its 1-skeleton as a k-coloured graph where , , inherited from and defined by if and only if where are the canonical generators for . The factorisation property in for elements of degree where gives rise to relations between the edges of .
As with graph-algebras one may associate a C*-algebra to a k=graph:
- if ;
- are mutually orthogonal projections;
- if then ;
- for all and .
is then the universal C*-algebra generated by a Cuntz–Krieger -family.
- Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics 103, American Mathematical Society