Let $S$ be a set of $n$ elements $\{1,2,\dots,n\}$ 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 $(R\_1\setminus R\_2) ∪ (R\_2\setminus R\_1)$

Every vertex in $G$ has the same degree. What is the degree of a vertex in $G$?

How many connected components does $G$ have?

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My take-I am able to solve it only by taking small example like take $n=2$ and $n=3$ and not able to understand the solution in formal way.

Please, explain how do I solve such question(s) formally.

Any help is highly appreciated in advance.