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：无向图的覆盖被用于分布式计算，有向图的分拆则应用于程序语义学。我们从图论角度研究这两个概念以凸显其相似性——二者均通过满射图同态定义。特别地，通用覆盖与完全分拆是无限树，当初始图有限时它们具有正则性。正则性指一棵树的非同构子树仅有有限多个。Leighton 和 Norris 已为覆盖建立了两条重要定理，我们为分拆证明了类似结论。在研究 Leighton 定理的艰深证明过程中，我们通过为被覆盖或分拆图的边附加有限或无限权重，实现了覆盖与分拆的统一推广。该推广为任意有限图的通用覆盖提供了规范分解——该分解在未使用权重时（可证明）不存在。引入无限权重使我们能对具有可数无穷度节点的正则树进行有限描述。我们还将特征多项式的经典分解定理推广至加权图及其覆盖情形。