Aperiodic graph

In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G. In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G. In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every k divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by n, the length of that cycle. Suppose that G is strongly connected and that k divides the lengths of all cycles in G.Consider the results of performing a depth-first search of G, starting at any vertex, and assigning each vertex v to a set Vi where i is the length (taken mod k) of the path in the depth-first search tree from the root to v. It can be shown (Jarvis & Shier 1996) that this partition into sets Vi has the property that each edge in the graph goes from a set Vi to another set V(i + 1) mod k. Conversely, if a partition with this property exists for a strongly connected graph G, k must divide the lengths of all cycles in G. Thus, we may find the period of a strongly connected graph G by the following steps: