Some results on the lexicographic product of vertex-transitive graphs

Many large graphs can be constructed from existing smaller graphs by using graph operations, for example, the Cartesian product and the lexicographic product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the lexicographic products of vertex-transitive and of edge-transitive graphs. In particular, we show that the lexicographic product of Cayley graphs is a Cayley graph.