Each can be embedded in the orientable surface of genus 3, in which they form dual graphs.

This is a 3-regular (cubic) graph with 56 vertices and 84 edges, named after Felix Klein.

It has book thickness 3 and queue number 2.

[1] It can be embedded in the genus-3 orientable surface (which can be represented as the Klein quartic), where it forms the Klein map with 24 heptagonal faces, Schläfli symbol {7,3}8.

According to the Foster census, the Klein graph, referenced as F056B, is the only cubic symmetric graph on 56 vertices which is not bipartite.

The characteristic polynomial of this 56-vertex Klein graph is equal to

This is a 7-regular graph with 24 vertices and 84 edges, named after Felix Klein.

It can be embedded in the genus-3 orientable surface, where it forms the dual of the Klein map, with 56 triangular faces, Schläfli symbol {3,7}8.

[4] It is the unique distance-regular graph with intersection array

[5] The automorphism group of the 7-valent Klein graph is the same group of order 336 as for the cubic Klein map, likewise acting transitively on its half-edges.

The characteristic polynomial of this 24-vertices Klein graph is equal to