Zero-symmetric graph

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

In the mathematical field of graph theory, a zero-symmetric graph is a vertex-transitive cubic graph whose edges are partitioned into three different equivalence classes (orbits) by the symmetry group of the graph. In these graphs, for every two vertices u and v, there is exactly one graph automorphism that takes u into v.

The smallest zero-symmetric graph is a nonplanar graph with 18 vertices; its LCF notation is [5,−5]⁹. Among planar graphs, the graphs of the truncated cuboctahedron and truncated icosidodecahedron are also zero-symmetric.

The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.

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)