Some recent discoveries about half-arc-transitive graphsConder, Marston D. E. (Avtor) Potočnik, Primož (Avtor) Šparl, Primož (Avtor) graphedge-transitivevertex-transitivearc-transitivehalf arc-transitiveWe present some new discoveries about graphs that are half-arc-transitive (that is, vertex- and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser ▫$D_4$▫ (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser of order 16 that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.20152021-12-31 00:38:13Neznano17613sl