Neighbourhood Graphs of Cayley Graphs for Finitely-generated Groups

In this short note the neighbourhood graph of a Cayley graph is considered. It has, as nodes, a symmetric generating set of a finitely-generated group Γ. Two nodes are connected by an edge if one is obtained from the other by multiplication on the right by one of the generators. Two necessary conditions on the graphs are shown. One is a condition on the degrees of the graph, the other concerns complete subgraphs.