# The k-point exponent set of primitive digraphs with girth 2

The k-point exponent set of primitive digraphs with girth 2 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# The k-point exponent set of primitive digraphs with girth 2

, Volume 24 (4) – Oct 12, 2008
10 pages      /lp/springer-journals/the-k-point-exponent-set-of-primitive-digraphs-with-girth-2-gmtoxiK0iA
Publisher
Springer Journals
Copyright © 2008 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-005-5118-7
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 12, 2008

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