The Distinguishing Numbers and the Distinguishing Indexes ofCayley Graphs
2021
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 $$2$$
.
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