Science in China Series A: Mathematics

, Volume 42, Issue 3, pp 264–271

# Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

• Zhang Fuji
• Yong Xuerong
Article

## Abstract

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let$$C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)$$ be a directed circulant graph. Let$$\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)$$ and$$\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)$$ be the numbers of spanning trees and of Eulerian trails, respectively. Then
$$\begin{array}{*{20}c} \begin{gathered} \lim \frac{1}{k}\sqrt[p]{{T\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \lim \frac{1}{{k!}}\sqrt[p]{{E\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \end{gathered} & {p \to \infty .} \\ \end{array}$$
Furthermore, their line digraph and iterations are dealt with and similar results are obtained for undirected circulant graphs.

## Keywords

asymptotic enumeration Eulerian trails graph circulant

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© Science in China Press 1999

## Authors and Affiliations

• Zhang Fuji
• 1
• Yong Xuerong
• 2
• 3
1. 1.Department of MathematicsXiamen UniversityXiamenChina
2. 2.Department of Computer ScienceHong Kong University of Science & TechnologyHong KongChina
3. 3.Department of MathematicsXinjiang UniversityUrumqiChina