Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings. We prove similar statements for unfoldings. Our study of the difficult proof of Leighton's Theorem lead us to generalize coverings and similarly, unfoldings, by attaching finite or infinite weights to edges of the covered or unfolded graphs. This generalization yields a canonical factorization of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing infinite weights provides us with finite descriptions of regular trees having nodes of countably infinite degree. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

翻译：无方向图形的覆盖面用于分布式计算, 以及程序语义中定向图形的展出。 我们从图表理论角度研究这两个概念, 以突出它们的相似性。 我们从图表理论角度研究这两个概念, 以突出它们之间的相似性, 因为这两个概念都是用想象式图形同质体来定义的。 特别是, 通用覆盖面和完整的展示面是无限的树木, 如果初始图形是有限的, 则这些树木是正常的。 规律性意味着一棵树有数量有限的许多次树, 直至非形态化。 由Leighton 和 Norris 建立了两个重要的理论来覆盖。 我们用相似的语句来证明它们的发展过程。 我们对Leighton 理论的艰难证明让我们对覆盖面进行概括化, 类似地, 通过将有限或无限的重量附加在覆盖面或展出图的边缘。 这种概括性能产生一个简单系数, 任何有限图形( 可能) 并不存在不使用重量。 引入无限的重量向我们提供了常规树的限定性描述, 具有可计量的无限的无限度度度。 我们对常规树的描述。 我们对Leightononalno loginalizalalal graphal gration gration gration graducalmentalmentalmentalmentalmentalmentalmentals) acus。