Zero-symmetric graph

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

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

Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric.

These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.

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.

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