Finite automorphism group of a graph

Question

Let $G$ be a simple graph and $Aut(G)$ denotes the automorphism group. Then prove or disprove,

Suppose $Aut(G)$ and $diam(G)$ are finite for a graph $G$. Then $|V(G)|$ is finite.

I know $Aut(G)$ is finite need not imply that $|V(G)|$ is finite, for example, Infinite path. But will the diameter condition make any changes?

Answer 1

Here's an example of a graph with an infinite number of vertices, finite automorphism group, and finite diameter.

We construct an infinite path centered at the vertex $v_0$. That is, we have a path of vertices $(\dots , v_{-2}, v_{-1}, v_0, v_1, v_2, \dots)$. Next we add an vertex $u$ such that $u$ is adjacent to each $v_i$. The inclusion of $u$ ensures that the diameter of the graph is finite. Now we add a final vertex $w$ such that $w$ is adjacent to $v_0$. Any automorphism must map $v_0$ to itself since $v_0$ is the only vertex with degree 4. Hence, the only automorphism is is the map that sends $v_i$ to $v_{-i}$ and leaves $v_0, u, w$ fixed.