Affiliation:
1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
2. Novosibirsk State University
Abstract
The purpose of this survey is to describe invariants of cyclic coverings of graphs. The covered graph is assumed to be fixed, and the cyclic covering group has an arbitrarily large order. A classical example of such a covering is a circulant graph. It covers a one-vertex graph with a prescribed number of loops. More sophisticated objects representing the family of cyclic coverings include $I$-, $Y$-, and $H$-graphs, generalized Petersen graphs, sandwich-graphs, discrete tori, and many others. We present analytic formulae for counting rooted spanning forests and trees in cyclic coverings, establish their asymptotics, and study the arithmetic properties of these quantities. Moreover, in the case of circulant graphs we give exact formulae for computing the Kirchhoff index and present structural theorems for the Jacobians of such graphs.
Bibliography: 95 titles.
Funder
Ministry of Science and Higher Education of the Russian Federation
Publisher
Steklov Mathematical Institute
Cited by
1 articles.
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