A voltage graph is a graph together with an assignment of elements of a group 
, called the voltage
group, to the oriented edges of the graph. More
precisely, if each edge of a graph 
is replaced by two oppositely oriented darts 
and 
, a voltage assignment is a function 
from the darts to 
such that
 |
(1)
|
A voltage assignment may be specified by choosing one orientation of each edge and assigning that dart a voltage. The inverse rule then
determines the voltage on the oppositely oriented dart.
When a small base graph is used explicitly, a chosen
dart from vertex 
to vertex 
with voltage 
is often recorded as a voltage triple 
.
The voltage assignment determines a derived graph 
whose vertex
set is 
.
For an oriented edge 
of 
with voltage 
and for each 
, the derived graph
has an edge
 |
(2)
|
Thus the product of voltages along a walk in the base graph records how the lifted walk moves among the sheets of the
cover. The natural projection that sends 
to 
gives a covering graph projection.
The voltage group acts on 
by left multiplication on the second coordinate, giving
a semiregular group action whose orbits
are the fibers of the projection,
that is, the sets of vertices mapping to single vertices of 
.
In this context, a graph cover of 
means a graph 
together with a graph projection 
such that, at each vertex
of 
, the edges incident to 
are mapped bijectively onto
the edges incident to 
.
Thus, locally around each lifted vertex, the cover
looks exactly like the base graph.
For example, the Petersen graph is the derived graph of a voltage graph whose voltage group
is the cyclic group 
, which arises from the inherent symmetry of the graph. To
recover a voltage presentation from a given graph, examine the structure of its automorphism group to identify subgroups
whose action on the vertices is a semiregular
group action. When the desired presentation has a cyclic
group as its voltage group, the relevant subgroups
are cyclic. In the conventional Petersen
graph embedding, rotation
by 
generates such a subgroup,
which is isomorphic to 
.
Since no nonidentity rotation fixes a vertex, this subgroup acts semiregularly
on the vertices and is used as the voltage group.
The group orbits of the 
subgroup are the five outer-cycle
vertices and the five inner-star vertices. The corresponding base
graph has one vertex for each group orbit, denoted
and 
, respectively.
The voltage in each triple records how far the lifted edge shifts the second coordinate in 
. In this embedding, an outer edge
has shift 1, a spoke has shift 0, and an inner-star edge has shift 2. With these
choices, one convenient set of voltage triples is
 |
(3)
|
Using the base-graph vertex labels 1 and 2 for 
and 
, respectively, gives the numeric voltage triples
 |
(4)
|
where the third entry in each triple is interpreted as a residue modulo 5. Writing subscripts modulo 5, the derived graph
has vertices 
and 
and edges
 |
(5)
|
These three edge types are the outer 5-cycle, the five spokes, and the inner pentagram in this embedding.
Ordinary voltage assignments provide a compact way to describe regular graph covers. More generally, permutation voltage assignments generate arbitrary graph covers (Gross and Tucker 1977). Voltage graphs are especially useful in topological graph theory, where they encode graph covers and derived embeddings on surfaces (Gross 1974, Gross and Tucker 1987).
