# Zero-symmetric graph

In the mathematical field of graph theory, a **zero-symmetric graph** is a 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]

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 |

The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.[2] These graphs are nowadays referred to as cubic GRR (Graphical Regular Representations). [3]

## Examples

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

Among planar graphs, the truncated cuboctahedral and 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 Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of 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

- Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981),
*Zero-symmetric graphs*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, ISBN 0-12-194580-4, MR 0658666 - Coxeter, Frucht & Powers (1981), p. ix.
- Lauri, Josef; Scapellato, Raffaele (2003),
*Topics in Graph Automorphisms and Reconstruction*, London Mathematical Society Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037. - Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.
- Coxeter, Frucht & Powers (1981), pp. 75 and 80.
- Coxeter, Frucht & Powers (1981), p. 55.
- Conder, Marston D. E.; Pisanski, Tomaž; Žitnik, Arjana (2017), "Vertex-transitive graphs and their arc-types",
*Ars Mathematica Contemporanea*,**12**(2): 383–413, arXiv:1505.02029, doi:10.26493/1855-3974.1146.f96, MR 3646702 - Potočnik, Primož; Spiga, Pablo; Verret, Gabriel (2013), "Cubic vertex-transitive graphs on up to 1280 vertices",
*Journal of Symbolic Computation*,**50**: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, MR 2996891. - Coxeter, Frucht & Powers (1981), p. 10.