Unfoldings and coverings of weighted graphs

TL;DR

This work unifies unfoldings and coverings of weighted graphs by treating both as locally bijective graph homomorphisms under different neighborhood notions, and extends them to weights drawn from $\mathbb{N}_{+}\cup\{\omega\}$. It proves weighted generalizations of Norris and Leighton theorems, showing that universal coverings and complete unfoldings of finite weighted graphs are regular, and that weighted finite graphs admit canonical finite descriptions whose universal coverings yield strongly regular trees. The paper also develops a polynomial-factorization theory for coverings via weight matrices and establishes constructive criteria for the existence of common coverings, broadening algebraic graph theory with the notion of strongly regular trees. Overall, it provides a coherent framework for analyzing infinite yet finitely describable structures arising from weighted graph unfoldings and coverings, with clear implications for automated reasoning about network semantics and spectral properties.

Abstract

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.

Figures (11)

• Figure 1: A weighted surjection, see Example 2.2(1).
• Figure 2: The weighted set S of Example 2.2(3).
• Figure 3: A digraph $G$ and its quotient $G/\approx$, cf. Example 3.12.
• Figure 4: None of these graphs covers any smaller graph. See Example 4.4.
• Figure 5: See Remark 4.17