Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree,[2] so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree.[3] The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.[4][5]
^Gross, J.L.; Yellen, J. (2004). Handbook of Graph Theory. CRC Press. p. 491. ISBN1-58488-090-2.