Bipartite Double Graph
The bipartite double graph, also called the Kronecker cover, Kronecker double cover, bipartite double cover, canonical double cover, or bipartite double, of a given graph 
is constructed by making two copies of the vertex set of 
(omitting the initial edge set entirely) and constructing edges 
and 
for every edge 
of 
. The bipartite double graph is equivalent to the graph categorical product 
.
In a non-bipartite connected graph, exactly one double cover is bipartite. However, a bipartite or disconnected graph may have more than one bipartite double graph, leading Pisanski (2018) to suggest than one of the alternate names should be used for this concept.
Note that the bipartite double differs from the plain double graph in that the initial edge set is discarded in the bipartite double graph, while it is retained in the double graph.
The following table summarizes bipartite double graphs for some named graphs and classes of graphs.
 |
bipartite double of  |
| 16-cell graph |
Haar graph  |
| 4-antiprism graph |
quartic vertex-transitive graph Qt48 |
| 5-antiprism graph |
Haar graph  |
| Biggs-Smith graph |
cubic symmetric graph  |
| Clebsch graph |
hypercube graph  |
complete graph  |
crown graph  |
| Coxeter graph |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
cubic symmetric graph  |
| cubic vertex-transitive graph Ct41 |
great rhombicuboctahedral graph |
cubical graph  |
 |
| cuboctahedral graph |
rolling cube graph |
cycle graph  |
cycle graph  for  odd;  for  even |
dodecahedral graph  |
cubic symmetric graph  |
| Doyle graph |
 -Bouwer graph |
Dürer graph  |
cubic vertex-transitive graph Ct38 |
empty graph  |
empty graph  |
 -folded cube graph for  |
hypercube graph  |
generalized Petersen graph  |
cubic vertex-transitive graph Ct38 |
generalized quadrangle  |
quartic vertex-transitive graph Qt66 |
Kneser graph  |
bipartite Kneser graph  |
| Kummer graph |
hypercube graph  |
ladder graph  |
 |
ladder rung graph  |
ladder rung graph  |
| 3-matchstick graph |
8-crossed prism graph |
Möbius ladder  for  |
prism graph  |
net graph  |
sunlet graph  |
odd graph  |
bipartite Kneser graph  |
path graph  |
 |
pentatope graph  |
crown graph  |
Petersen graph  |
Desargues graph  |
prism graph  |
prism graph  for  odd;  for  even |
| quartic vertex-transitive graph Qt45 |
torus grid graph  |
| quartic vertex-transitive graph Qt65 |
torus grid graph  |
rook graph  |
tesseract graph  |
rook graph  |
Kummer graph |
| Shrikhande graph |
Kummer graph |
square graph  |
 |
sunlet graph  |
sunlet graph  |
tesseract graph  |
 |
tetrahedral graph  |
cubical graph  |
transposition graph  |
 |
triangle graph  |
cycle graph  |
| truncated tetrahedral graph |
Nauru graph  |
utility graph  |
 |
Wagner graph  |
prism graph  |
web graph  |
web graph  for  odd;  for  even |
REFERENCES
Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, pp. 17 and 24, 1989.
DistanceRegular.org. "Bipartite Doubles." http://www.distanceregular.org/indexes/bipartitedoubles.html.Pisanski, T. "Not Every Bipartite Double Cover Is Canonical." Bull. ICA 82, 51-55, 2018.
