Loupekine snarks

Loupekine snark (first)
The first Loupekine snark
Vertices	22
Edges	33
Radius	3
Diameter	4
Girth	5
Chromatic number	3
Chromatic index	4
Properties	not planar
Table of graphs and parameters

Loupekine snark (second)
The second Loupekine snark
Vertices	22
Edges	33
Radius	3
Diameter	4
Girth	5
Chromatic number	3
Chromatic index	4
Properties	not planar
Table of graphs and parameters

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

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

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 ${\displaystyle D_{6}}$ (identified as [12,4] in the Small Groups Database). The orbits under the action of ${\displaystyle D_{6}}$ 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:

${\displaystyle \chi _{1}=(x-3)(x+2)^{3}(x^{3}+x^{2}-4x-2)(x^{3}-2x^{2}-x+1)^{2}\cdot (x-2)(x^{2}+2x-2)(x^{3}-3x+1)^{2}}$

and

${\displaystyle \chi _{2}=(x-3)(x+2)^{3}(x^{3}+x^{2}-4x-2)(x^{3}-2x^{2}-x+1)^{2}\cdot x(x^{2}-2)(x^{3}-5x+3)^{2}}$

References

1. http://doc.sagemath.org/pdf/en/reference/graphs/graphs.pdf ^[bare URL PDF]