Covering graph

The combinatorial formulation of covering graphs is immediately generalized to the case of multigraphs.

Then f is a covering map from C to G if for each v ∈ V2, the restriction of f to the neighbourhood of v is a bijection onto the neighbourhood of f(v) in G. Put otherwise, f maps edges incident to v one-to-one onto edges incident to f(v).

[3] This is an instance of the more general universal cover concept from topology; the topological requirement that a universal cover be simply connected translates in graph-theoretic terms to a requirement that it be acyclic and connected; that is, a tree.

An infinite-fold abelian covering graph of a finite (multi)graph is called a topological crystal, an abstraction of crystal structures.

This view combined with the idea of "standard realizations" turns out to be useful in a systematic design of (hypothetical) crystals.

The universal cover can be seen in this way as a derived graph of a voltage graph in which the edges of a spanning tree of the graph are labeled by the identity element of the group, and each remaining pair of darts is labeled by a distinct generating element of a free group.