In the mathematical field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, the Klein graphs are two different but related regular graphs, each with 84 edges. Each can be embedded in the orientable surface of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

3, in which they form dual graphs.

The cubic Klein graph

This is a 3- regular (

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

) graph with 56 vertices and 84 edges, named after Felix Klein. It is

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

, has chromatic number 3, chromatic index 3, radius 6, diameter 6 and

girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...

7. It is also a 3- vertex-connected and a 3- edge-connected graph. It has book thickness 3 and queue number 2. It can be embedded in the

-3 orientable surface (which can be represented as the Klein quartic), where it forms the Klein map with 24 heptagonal faces,

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

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 Bipartite may refer to: * 2 (number) * Bipartite (theology), a philosophical term describing the human duality of body and soul * Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...

. It can be derived from the 28-vertex Coxeter graph.

Algebraic properties

The automorphism group of the Klein graph is the group PGL₂(7) of order 336, which has PSL₂(7) as a normal subgroup. This group acts transitively on its half-edges, so the Klein graph is a symmetric graph. The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

of this 56-vertex Klein graph is equal to

x^7 \,(x-3)\,(x+2)^6 \left(x^2-2\right)^6 \left(x^2+x-4\right)^7 \left(x^2-2 x-1\right)^8

The 7-regular Klein graph

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

, has chromatic number 4, chromatic index 7, radius 3, diameter 3 and

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

₈. It is the unique distance-regular graph with intersection array

\

; however, it is not a distance-transitive graph.

Algebraic properties

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

of this 24-vertices Klein graph is equal to

(x-7) (x+1)^7 (x^2-7)^8

References

{{reflist Individual graphs Regular graphs