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

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.