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：无向图的覆盖用于分布式计算，而有向图的展开则用于程序语义学。我们从图论视角研究这两个概念以突出其相似性——它们均通过满射图同态来定义。特别地，通用覆盖和完全展开是无限树，当初始图有限时，这些树具有正则性。正则性意味着树仅有有限个互不同构的子树。Leighton和Norris已针对覆盖建立了两个重要定理。我们证明了展开的类似结论。对Leighton定理困难证明的研究促使我们通过为被覆盖或展开图的边附加有限或无限权重，将覆盖及展开概念进行推广。这种推广为任意有限图的通用覆盖提供了典范分解，而（可证明）不使用权重时该分解不存在。引入无限权重使我们得以用有限方式描述节点具有可数无穷度的正则树。我们还将其特征多项式的经典分解定理推广至加权图及其覆盖。