When considering the Cayley graph of $\mathbb{Z}_3 \times \mathbb{Z}_3$ with respect to the generating set $S=\{(0,1),(1,0)\}$, as shown in Figure (1) below, can I regard the graph in Figure (2) as one of its automorphisms?
2026-03-25 16:00:14.1774454414
Automorphism of a Cayley graph
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If you mean to ask if your second graph represents the Cayley graph of your group with respect to the generating set $\{(0,1),(1,1)\}$ corresponding to the automorphism of $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ given by
$$(0,1)\mapsto (0,1) \quad(1,0)\mapsto (1,1)$$
then yes it does.