On Automorphism Groups of Deleted Wreath Products

Let \(\Gamma _1\) and \(\Gamma _2\) be digraphs. The deleted wreath product of \(\Gamma _1\) and \(\Gamma _2\), denoted \(\Gamma _1\wr _d\Gamma _2\), is the digraph with vertex set \(V(\Gamma _1)\times V(\Gamma _2)\) and arc set \(\{((x_1,y_1),(x_2,y_2)):(x_1,x_2)\in A(\Gamma _1)\mathrm{\ and\ }y_1\not = y_2\mathrm{\ or\ }x_1 = x_2\mathrm{\ and\ }(y_1,y_2)\in A(\Gamma _2)\}\). We study the automorphism group of \(\Gamma _1\wr _d\Gamma _2\), which always contains a natural subgroup isomorphic to \({\mathrm{Aut}}(\Gamma _1)\times {\mathrm{Aut}}(\Gamma _2)\). In particular, we focus on the characterization of digraph pairs \(\Gamma _1\), \(\Gamma _2\) such that \(\Gamma _1 \wr _d \Gamma _2\) admits automorphisms not contained in this natural subgroup of \({\mathrm{Aut}}(\Gamma _1 \wr _d \Gamma _2)\). We provide methods to construct such pairs of digraphs, and also give several sufficient conditions under which no such additional automorphisms exist. As a corollary of our results, we provide a method for constructing new half-arc-transitive graphs from known ones using deleted wreath products.