Exponents of 2-regular digraphs

AbstractA digraph G is called primitive if for some positive integer k, there is a walk of length exactly k from each vertex u to each vertex v (possibly u again). If G is primitive, the smallest such k is called the exponent of G, denoted by exp(G). A digraph G is said to be r-regular if each vertex in G has outdegree and indegree exactly r.It is proved that if G is a primitive 2-regular digraph with n vertices, then exp(G)⩽(n−1)2/4+1. Also all 2-regular digraphs with exponents attaining the bound are characterized. This supports a conjecture made by Shen and Greegory

Lower bounds for the length of reset words in eulerian automata

For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2-3n+4/2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. © 2013 World Scientific Publishing Company

Lower Bounds for the Length of Reset words in Eulerian Automata

For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2-3n+4/2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. © 2011 Springer-Verlag.Supported by the Russian Foundation for Basic Research, grant 10-01-00524, and by the Federal Education Agency of Russia, grant 2.1.1/13995