For C*-algebra 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 . Aside from its category theory definition, one can think of k-graphs as a higher-dimensional analogue of directed graphs (digraphs). k- here signifies the number of "colors" of edges that are involved in the graph. If k=1, a k-graph is just an ordinary directed graph. If k=2, there are two different colors of edges involved in the graph and additional factorization rules of 2-color equivalent classes should be defined. The factorization rule on k-graph skeleton is what distinguishes one k-graph defined on the same skeleton from another k-graph. k can be any natural number greater than or equal to 1. The reason k-graphs were first introduced by Kumjian, Pask et al. was to create examples of C*-algebras from them. k-graphs consist of two parts: skeleton and factorization rules defined on the given skeleton. Once k-graph is well-defined, one can define functions called 2-cocycles on each graph, and C*-algebras can be built from k-graphs and 2-cocycles. k-graphs are relatively simple to understand from a graph theory perspective, yet just complicated enough to reveal different interesting properties at the C*-algebra level. The properties such as homotopy and cohomology on the 2-cocycles defined on k-graphs have implications to C*-algebra and K-theory research efforts. No other known use of k-graphs exist to this day; k-graphs are studied solely for the purpose of creating C*-algebras from them. Background The finite graph theory in a directed graph form a category under concatenation called the free object category (generated by the graph). The length of a path in gives a functor from this category into the natural numbers . A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask. Examples 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 by is a k-graph. Let , then is a k-graph when gifted with the structure maps , , and . Notation The notation for k-graphs is borrowed extensively from the corresponding notation for categories: For let . By the factorisation property it