Is there a vertex-transitive graph with $n$ automorphisms that has $n \geq 3$? I've established that a lower bound for the automorphism group on a vertex-transitive undirected graph $G$ is $n$ using the orbit-stabilizer theorem. I'm now curious whether it's possible to achieve that bound nontrivially - I have boring examples for $n = 1$ and $n = 2$, but I'd like something more interesting.
 A: This is wrong as noted in comments, I'll leave it here as a failed attempt.
Its impossible for all $n\geq 3$.
Notice that the only transitive subgroups of $S_n$ are those generated by $n$-cycles.
Therefore we can label our vertices $v_1,v_2\dots v_n$ in such a way that the only $n$ automorphisms are those of the form $f_i$ such that $f_i(v_j)=v_{i+j}$ (working $\bmod n$ of course).
We conclude that the graph can be formed in the following way.
Pick a subset $A$ of $\{1,2\dots n-1\}$ and connect $A_i$ to $A_j$ if and only if $i-j\in A$.
But notice that the graph is undirected, so not every selection gives us a correct graph(in the case in which we allow for directed graphs taking $A=\{1\}$ gives us a transitive graph with $n$ automorphisms (the directed cycle).
If you want to get an undirected graph you need for $A$ to satisfy the following condition: $a\in A \iff n-a\in A$.
Notice that by doing all of the automorphisms of $D_{2n}$ come into play, so for example the map that fixes $v_1$ and sends every other $v_j$ to $v_{-j}$ is also an automorphism.
A: Yes, there are larger graphs for which the bound is achieved.  Let $X$ be the Cayley graph of $S_7$ generated by a set $S$ of transpositions for which the transposition graph $T(S)$ is an asymmetric tree.  Then $X$ is a vertex-transitive graph on $7!=5,040$ vertices and it has exactly $5,040$ automorphisms.  
More generally, (Godsil and Royle, Algebraic Graph Theory) proved that if $S$ is a set of transpositions generating $S_n$ and the transposition graph of $S$ is an asymmetric tree, then the Cayley graph of $S_n$ with respect to generating set $S$ has exactly $|S_n|$ automorphisms.  Such graphs are called graphical regular representations (GRRs) and have the smallest possible full automorphism group in some sense.
