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In mathematics , a k-graph -LRB- or higher-rank graph , graph of rank k -RRB- 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 .. mathematics. mathematics. graph. graph theory. countable. countable. category. category ( mathematics ). domain. domain of a function. codomain. codomain. functor. functor. factorisation. factorisation. Aside from its category theory definition , one can think of k-graphs as higher dimensional analogue of directed graphs -LRB- digraphs -RRB- .. category. category ( mathematics ). k - here signifies the number of `` colors '' of edges that are involved in the graph .. graph. graph theory. If k = 1 , k-graph is just a regular directed graph .. graph. graph theory. 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 .. graph. graph theory. The factorization rule on k-graph skeleton is what distinguishes one k-graph defined on the same skeleton from another k-graph .. graph. graph theory. 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 * - algebra 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 .. graph. graph theory. k-graphs are relatively simple to understand from graph theory perspective , yet just complicated enough to reveal different interesting properties in the C * - algebra level .. graph. graph theory. 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 .