Zero-symmetric graph

Source: Wikipedia, the free encyclopedia.
18-vertex zero-symmetric graph
The smallest zero-symmetric graph, with 18 vertices and 27 edges
Truncated cuboctahedron
The truncated cuboctahedron, a zero-symmetric polyhedron
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive 		→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric

In the

connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.[1]

The smallest zero-symmetric graph with two edge orbits

The name for this class of graphs was coined by

R. M. Foster in a 1966 letter to H. S. M. Coxeter.[2] In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.[3]

Examples

The smallest zero-symmetric graph is a nonplanar graph with 18 vertices.[4] Its LCF notation is [5,−5]9.

Among

truncated icosidodecahedral graphs are also zero-symmetric.[5]

These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.[6]

These examples also have three different symmetry classes (orbits) of edges. However, there exist zero-symmetric graphs with only two orbits of edges. The smallest such graph has 20 vertices, with LCF notation [6,6,-6,-6]5.[7]

Properties

Every finite zero-symmetric graph is a

combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices.[8]

Unsolved problem in mathematics:

Does every finite zero-symmetric graph contain a

Hamiltonian cycle
?

(more unsolved problems in mathematics)

All known finite connected zero-symmetric graphs contain a

Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian.[9] This is a special case of the Lovász conjecture
that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian.

See also

  • Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the definition of zero-symmetric graphs)

References

  1. MR 0658666
  2. ^ Coxeter, Frucht & Powers (1981), p. ix.
  3. ISBN 9780521529037
    .
  4. ^ Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.
  5. ^ Coxeter, Frucht & Powers (1981), pp. 75 and 80.
  6. ^ Coxeter, Frucht & Powers (1981), p. 55.
  7. MR 3646702
  8. MR 2996891
    .
  9. ^ Coxeter, Frucht & Powers (1981), p. 10.
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