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.

The smallest zero-symmetric graph, with 18 vertices and 27 edges
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

The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter. In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.

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