Abstract
A digraph is primitive provided there is a positive integer k such that for each pair of vertices u and v there exist walks of length k from u to v and from v to u. The scrambling index of a primitive digraph D is the smallest positive integer k such that for each pair of vertices u and v in D there is a vertex w such that there exist walks of length k from u to w and from v to w . A two-colored digraph is a digraph each of whose arc is colored by red or blue. In this paper we generalize the notion of scrambling index of a primitive digraph to that of two-colored digraph. We define the scrambling index of a two-colored digraph D (2) to be the smallest positive integer h+ l over all pairs of nonnegative integers (h, l) such that for each pair of distinct vertices u and v there is a vertex w with the property that there are walks form u to w and from v to w consisting of h red arcs and l blue arcs. For two-colored Wielandt digraph on n 4 vertices we show the scrambling index lies on the interval (n 2 3n + 3, n 2 2n + 2).
Highlights
By a nonnegative integer vector x ≥ 0 we meant a vector each of whose entry is a nonnegative integer
A walk of length k from u to v is a sequence of arcs of the form u = v0 → v1, v1 → v2, . . . , vk−1 → vk = v
For a primitive two-colored digraph D(2) we define the scrambling index of D(2) to be the smallest positive integer h + l over all nonnegative integers h and l such that for every pair of vertices u and v in D(2) there is a (h,l) vertex w with the property that there is a u −→ w walk and a v (−h→,l) w walk
Summary
By a nonnegative integer vector x ≥ 0 we meant a vector each of whose entry is a nonnegative integer. For a primitive two-colored digraph D(2) we define the scrambling index of D(2) to be the smallest positive integer h + l over all nonnegative integers h and l such that for every pair of vertices u and v in D(2) there is a (h,l) vertex w with the property that there is a u −→ w walk and a v (−h→,l) w walk. An automaton A is synchronizing with reset word of length h + l if there exists a vertex u in D(2) such that for each vertex v in D(2) there is a v (−h→,l) u walk, the order of appearance of red and blue arcs in each v → u walk are the same. The scrambling index of a two-colored digraph may be used as a lower bound for the length of a reset word for a synchronizing automaton with two input letters.
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