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 an undirected graph in which all vertices are symmetric to each other, each vertex has exactly three incident edges, and these three edges are not symmetric to each other. More precisely, it is a vertex-transitive cubic graph whose edges are partitioned into three different orbits by the automorphism group. 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)