I was reading about Lovász Conjecture and came across the following definition on Wikipedia of a vertex-transitive graph (see below).
$\bullet$ It states that a graph is vertex-transitive if for any two vertices $u$ and $v$ of the graph, there is some automorphism (i.e. a relabeling of vertices of a graph) $f: V(G)\rightarrow V(G)$ where $f(u)=v$.
$\textbf{QUESTION:}$ I'm having a hard time figuring out how to use this definition in this context; so, my question is why are certain graphs vertex transitive and others not? For example, what is the function for the graph below that makes it vertex transitive?
Suppose you wanted to swap $v_1$ and $v_2$. Then, you could leave $v_3$ and $v_4$, and all of the connections in the graph would be the same (e.g. the new $v_1$ and the old $v_1$ are both still connected to the vertices labelled $v_2,v_3,v_4$). This graph is $K_4$ which is particularly nice in that any rearrangement preserves that property.
To see an example that wouldn't work, take a graph with two vertices of different degree. Then, no matter how you try to swap them you won't be able to get a graph with the same connectivity relationships.