Q6

Let $S$ be a set of $n$ elements $\left\{1, 2,\ldots, n\right\}$ and $G$ a graph with $2^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly $2$ elements. Note: The symmetric difference of two sets $R_{1}$ and $R_{2}$ is defined as $\left ({R_{1}}\setminus{R_{2}} \right)$ $\cup$ $\left ({R_{2}}\setminus{R_{1}} \right)$ Every vertex in $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?