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Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs. (English) Zbl 0929.05018

The authors established asymptotic formulas for the number of spanning trees and the number of Eulerian trails in circulant digraphs and in circulant graphs. In particular, if \(T\) is the number of spanning trees and \(E\) is the number of Eulerian trails in any circulant digraph with \(p\) vertices and out-degree \(k\) then it is shown that \({1\over k}[T]^{{1\over p}}\sim 1\) and \({1\over k!}[E]^{{1\over p}}\sim 1\) for \(p\to\infty\). Additional results are established for line graphs and for circulant graphs.

MSC:

05C05 Trees
05C30 Enumeration in graph theory
05A16 Asymptotic enumeration
05C45 Eulerian and Hamiltonian graphs
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