Zero-symmetric graph
![18-vertex zero-symmetric graph](http://upload.wikimedia.org/wikipedia/commons/thumb/3/36/18-vertex_zero-symmetric_graph.svg/200px-18-vertex_zero-symmetric_graph.svg.png)
![Truncated cuboctahedron](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Great_rhombicuboctahedron.png/200px-Great_rhombicuboctahedron.png)
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 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]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Two-edge-orbit_GRR.svg/180px-Two-edge-orbit_GRR.svg.png)
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 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]
Does every finite zero-symmetric graph contain a Hamiltonian cycle?
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, p. 66, 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.
- Algebraic graph theory
- Graph families
- Regular graphs