→
{\displaystyle {\boldsymbol {\rightarrow }}} distance-regular ←
{\displaystyle {\boldsymbol {\leftarrow }}} t-transitive, t ≥ 2 | ←
{\displaystyle {\boldsymbol {\leftarrow }}} strongly regular →
{\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive | |
 | ||
(if connected)
vertex- and edge-transitive →
{\displaystyle {\boldsymbol {\rightarrow }}} →
{\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive and regular | ||
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.
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.
References
Half-transitive graph Wikipedia(Text) CC BY-SA