Aperiodic graph

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 V_i where i is the length (taken modulo 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 V_i has the property that each edge in the graph goes from a set V_i 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:

• Perform a depth-first search of G
• For each e in G that connects a vertex on level i of the depth-first search tree to a vertex on level j, let k_e = j − i − 1.
• Compute the greatest common divisor of the set of numbers k_e.

The graph is aperiodic if and only if the period computed in this fashion is 1.

If G is not strongly connected, we may perform a similar computation in each strongly connected component of G, ignoring the edges that pass from one strongly connected component to another.

Jarvis and Shier describe a very similar algorithm using breadth first search in place of depth-first search; the advantage of depth-first search is that the strong connectivity analysis can be incorporated into the same search.

In a strongly connected graph, if one defines a Markov chain on the vertices, in which the probability of transitioning from v to w is nonzero if and only if there is an edge from v to w, then this chain is aperiodic if and only if the graph is aperiodic. A Markov chain in which all states are recurrent has a strongly connected state transition graph, and the Markov chain is aperiodic if and only if this graph is aperiodic. Thus, aperiodicity of graphs is a useful concept in analyzing the aperiodicity of Markov chains.

Aperiodicity is also an important necessary condition for solving the road coloring problem. According to the solution of this problem (Trahtman 2009), a strongly connected directed graph in which all vertices have the same outdegree has a synchronizable edge coloring if and only if it is aperiodic.

• Jarvis, J. P.; Shier, D. R. (1996), "Graph-theoretic analysis of finite Markov chains", in Shier, D. R.; Wallenius, K. T. (eds.), Applied Mathematical Modeling: A Multidisciplinary Approach (PDF), CRC Press.
• Trahtman, Avraham N. (2009), "The road coloring problem", Israel Journal of Mathematics, 172 (1): 51–60, arXiv:0709.0099, doi:10.1007/s11856-009-0062-5.