Let $S$ be a set of $n$ elements $\{1, 2, \dots n\}$ and $G$ a graph with $2^n$ vertices, where each vertex corresponds to a distinct subset of $S$. Two vertices are adjacent if the symmetric difference of the corresponding sets has exactly $2$ elements. Every vertex in $G$ has the same degree. What is the degree of a vertex in $G$ and how many connected component does $G$ have, respectively?
- $n, 3$
- $(n(n-1))/2,2$
- $1, n$
- $n+1, n$