### Citations

6 | Primitivity positive matrix pairs: algebraic characterization graph theoritic description and 2D systems interpretations
- Fornasini, Valcher
- 1998
(Show Context)
Citation Context ...e greatest common divisor of the determinants of the 2 by 2 submatrices of M , otherwise. The following theorem presents an algebraic characterization for a primitive two-colored digraph. Theorem 1.1 =-=[5]-=- Let D(2) be a two-colored digraph with at least one arc of each color. The two-colored digraph D(2) is primitive if and only if the content of its cycle matrix is 1. We generalize the notion of scram... |

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Primitive Digraphs with Large Exponents and Slowly Synchronizing Automata
- Ananichev, Volkov, et al.
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Citation Context ...vertices u and v in D(2) there is a vertex w with the property that there is a u (h,ℓ)! w walk and a v (h,ℓ)! w walk. The scrambling index ofD(2) is denoted by k(D(2)). Ananichev, Gusev, and Volkov =-=[3]-=- have used primitive digraphs with large exponents in attempt to find slowly synchronizing automata. Such primitive digraphs consist of cycles with two distinct lengths. An automaton on two input lett... |

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S.: Exponents of nonnegative matrix pairs
- Shader, Suwilo
- 2003
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Citation Context ... ! vn ! v1 and the (n 1)-cycle v1 ! v2 ! ! vn1 ! v1. As a consequence of Theorem 1.1 we have the following characterization for primitivity of a two-colored Wielandt digraph. Lemma 3.1 =-=[6]-=- A two-colored Wielandt digraph W (2)n on n vertices is primitive if and only if its cycle matrix M =[ r(C1) r(C2) b(C1) b(C2) ] = [ n 1 n 2 1 1 ] . Lemma 3.1 implies that a primitive two-colored Wi... |

3 | Primitive digraphs with the largest scrambling index
- Akelbek, Kirkland
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Citation Context ...raph consists of a Hamiltonian cycle v1 ! v2 ! ! vn ! v1 and the arc vn1 ! v1 as in Figure 1. The notion of scrambling index of a primitive digraph was first introduced by Akelbek and Kirkland =-=[1, 2]-=-. They define the scrambling index of a primitive digraph D to be the smallest positive integer k such that for every pair of vertices u and v in D there exists a vertex w in D such that there is a u ... |

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Coefficients of ergodicity and the scrambling index, Linear Algebra and its Applications
- Akelbek, Kirkland
(Show Context)
Citation Context ...raph consists of a Hamiltonian cycle v1 ! v2 ! ! vn ! v1 and the arc vn1 ! v1 as in Figure 1. The notion of scrambling index of a primitive digraph was first introduced by Akelbek and Kirkland =-=[1, 2]-=-. They define the scrambling index of a primitive digraph D to be the smallest positive integer k such that for every pair of vertices u and v in D there exists a vertex w in D such that there is a u ... |