The Distinguishing Numbers and the Distinguishing Indexes ofCayley Graphs

The distinguishing number (index) $$D(G)$$ ( $$D^{\prime }(G) $$ ) of a graph $$G $$ is the least integer $$d $$ such that $$G $$ has a vertex labeling (edge labeling) with $$d $$ labels which is preserved only by a trivial automorphism. In this paper, we investigate the distinguishing numbers and the distinguishing indexes of Cayley graphs. In particular, we obtain an upper bound for the distinguishing numbers of Cayley graphs. Also, we present a family of the Cayley graphs, graphical regular representations of a group, for each of which the distinguishing number as well as the distinguishing index is $$2$$ .