Some bounds of the generalized μ-scrambling indices of primitive digraphs with d loops

In 2010, Huang and Liu introduced a useful parameter called the generalized μ-scrambling indices of a primitive digraph. In this paper, we give some bounds for μ-scrambling indices of some primitive digraphs with d loops and the digraphs attained the sharp upper bounds are provided.

Reset Thresholds of Automata with Two Cycle Lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, [Formula: see text]. We establish reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length p and one simple cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length q and two simple cycles of length p. We obtain the reset thresholds of the form [Formula: see text], where [Formula: see text] is a linear function of p and q.