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
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.
