Let A be an n×n primitive Boolean matrix. γ(A) is the least number k such that A k=J. σ(A) is the number of 1 entry in A . In this paper, we consider the parameter M ′(k,n)= min {σ...Let A be an n×n primitive Boolean matrix. γ(A) is the least number k such that A k=J. σ(A) is the number of 1 entry in A . In this paper, we consider the parameter M ′(k,n)= min {σ(A)|A k=J, trace (A)=0} and obtain the values of M ′(2,n) and M ′(k,n) for k≥2n-6 . Furthermore, the characterization of solution of A 2=J with trace (A) =0 and σ(A)=3n-3 is completely determined.展开更多
Let Gn（C） be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and Gc（Jr~） the set of all primitive matrices in Gn（C）. In this paper, some necessary and sufficient condi...Let Gn（C） be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and Gc（Jr~） the set of all primitive matrices in Gn（C）. In this paper, some necessary and sufficient conditions for A in the semigroup Gn（C） to be primitive are given. We also show that Gc（Jn） is a subsemigroup of Gn（C）.展开更多
Let G =（Y, E） be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG（v）, is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vert...Let G =（Y, E） be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG（v）, is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vertices of G in such a way that expG（v1） ≤ expG（v2） ≤... ≤expG（vn）. Then expG（vk） is called the k-point exponent of G and is denoted by expG（k）, 1 〈 k 〈 n. We define the k-point exponent set E（n, k） ：= {expG（k）｜G= G（A） with A E CSP（n）｜, where CSP（n） is the set of all n × n central symmetric primitive matrices and G（A） is the associated graph of the matrix A. In this paper, we describe E（n, k） for all n, k with 1 〈 k 〈 n except n ≡ l（mod 2） and 1 〈 k 〈 n - 4. We also characterize the extremal graphs when k = 1.展开更多
This paper first establishes a distance inequality of the associated dia-graph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exp...This paper first establishes a distance inequality of the associated dia-graph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,…, n - 1}. There is no gap in it.展开更多
文摘Let A be an n×n primitive Boolean matrix. γ(A) is the least number k such that A k=J. σ(A) is the number of 1 entry in A . In this paper, we consider the parameter M ′(k,n)= min {σ(A)|A k=J, trace (A)=0} and obtain the values of M ′(2,n) and M ′(k,n) for k≥2n-6 . Furthermore, the characterization of solution of A 2=J with trace (A) =0 and σ(A)=3n-3 is completely determined.
基金Supported by the National Natural Science Foundation of China(11071272,11101087)the Postdoctoral Science Foundation of China(2013M531785)+1 种基金the Natural Science Foundation of Fujian Province(2013J05006)the Foundation of Fuzhou University(2012-XQ-30)
文摘Let Gn（C） be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and Gc（Jr~） the set of all primitive matrices in Gn（C）. In this paper, some necessary and sufficient conditions for A in the semigroup Gn（C） to be primitive are given. We also show that Gc（Jn） is a subsemigroup of Gn（C）.
基金Foundation item:The NSF(04JJ40002)of Hunan and the SRF of Hunan Provincial Education Department.
文摘Let G =（Y, E） be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG（v）, is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vertices of G in such a way that expG（v1） ≤ expG（v2） ≤... ≤expG（vn）. Then expG（vk） is called the k-point exponent of G and is denoted by expG（k）, 1 〈 k 〈 n. We define the k-point exponent set E（n, k） ：= {expG（k）｜G= G（A） with A E CSP（n）｜, where CSP（n） is the set of all n × n central symmetric primitive matrices and G（A） is the associated graph of the matrix A. In this paper, we describe E（n, k） for all n, k with 1 〈 k 〈 n except n ≡ l（mod 2） and 1 〈 k 〈 n - 4. We also characterize the extremal graphs when k = 1.
基金Foundation item:The Scientific Research Foundations of Hunan Provincial Education Department(02C448)Hunan University of Science and Technology(E50128)
文摘This paper first establishes a distance inequality of the associated dia-graph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,…, n - 1}. There is no gap in it.
