Half-transitive graph
In the mathematical field of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric. In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices.
| 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 | |
Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree, so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree. The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.
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