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Let D = (V,E) be a primitive digraph. The vertex exponent of D at a vertex v ∈ V, denoted by expD(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exp D (v 1) ≤ exp D (v 2) ≤ ... ≤ exp D (v n ). Then exp D (v k ) is called the k-point exponent of D and is denoted by exp D (k), 1 ≤ k ≤ n. In this paper we define e(n, k):=max{exp D (k)|D ∈ PD(n, 2)} and E(n, k):= {exp D (k)|D ∈ PD(n, 2)}, where PD(n, 2) is the set of all primitive digraphs of order n with girth 2. We completely determine e(n, k) and E(n, k) for all n, k with n ≥ 3 and 1 ≤ k ≤ n.
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Oct 12, 2008
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