$1$ Let $n2$ be a positive integer. Define the graph $G_n$ as follows: $V(G_n)$ is the set of all binary

strings on $n$ bits, with two vertices adjacent if they differ in exactly one bit. For example, in $G_4,$

$1100$ is adjacent to $0100$ but $1100$ is not adjacent to $0000.$

$($a$)$ Determine, with justification, $|V(G_n)|.$

$($b$)$ Determine, with justification, $|E(G_n)|.($Hint: Is $G_n$ a regular graph?$)$

$($c$)$ Prove $G_n$ is bipartite.