On automorphisms of direct products of Cayley graphs on abelian groups
Summary: Let \(X\) and \(Y\) be connected Cayley graphs on abelian groups, such that no two distinct vertices of \(X\) have exactly the same neighbours, and the same is true about \(Y\). We show that if the number of vertices of \(X\) is relatively prime to the number of vertices of \(Y\), then the direct product \(X \times Y\) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors \(X\) and \(Y)\). This was not previously known even in the special case where \(Y = K_2\) has only two vertices. The proof of this special case is short and elementary. The general case follows from the special case by standard arguments.
canonical bipartite double cover of a graph
scientific article; zbMATH DE number 7368134