Unfoldings and coverings of weighted graphs

TL;DR

This paper explores the concepts of coverings and unfoldings of weighted graphs, establishing new theorems and generalizations that connect these notions through weighted graph structures.

Contribution

It introduces a weighted generalization of coverings and unfoldings, providing canonical factorizations and extending classical theorems to weighted graphs.

Findings

Proved similar theorems for unfoldings as for coverings.

Generalized coverings and unfoldings using finite and infinite weights.

Established a factorization theorem for characteristic polynomials of weighted graphs.

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.