In the mathematical field of graph theory, the Loupekine snarks are two snarks, both with 22 vertices and 33 edges.

Loupekine snark (first)
The first Loupekine snark
Vertices22
Edges33
Radius3
Diameter4
Girth5
Chromatic number3
Chromatic index4
Propertiesnot planar
Table of graphs and parameters
Loupekine snark (second)
The second Loupekine snark
Vertices22
Edges33
Radius3
Diameter4
Girth5
Chromatic number3
Chromatic index4
Propertiesnot planar
Table of graphs and parameters

The first Loupekine snark graph can be described as follows (using the SageMath's syntax[1]):

lou1 = Graph({1:[2,3,4],
5:[6,10],6:[7],7:[8],8:[9],9:[10],
11:[16,12],12:[13],13:[14],14:[15],15:[16],
17:[2,5,16],18:[2,10,11], 19:[3,7,12],20:[3,6,13], 21:[9,4,14],22:[4,8,15]}).

The second Loupekine snark is obtained (up to an isomorphism) by replacing edges 5–6 and 11–12 by edges 5–12 and 6–11 in the first graph.

Properties

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Both snarks share the same invariants (as given in the boxes). The set of all the automorphisms of a graph is a group for the composition. For both Loupekine snarks, this group is the dihedral group   (identified as [12,4] in the Small Groups Database). The orbits under the action of   are :

1
2,3,4
17, 18, 19, 20, 21, 22
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

The characteristic polynomials are different, namely:

 

and

 

References

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