On Automorphism Groups of Deleted Wreath Products

Abstract

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.