$a)$ Let $Q_k$ be the graph whose vertex set corresponds to the set of all 01- sequences of length k, with an edge between two vertices if the sequences differ in exactly one place. Determine $d(Q_k)$ , $e(Q_k)$, $|Q_k|$.
$b)$ Show that $Q_k$ is bipartite.
Let us begin with $d(Q_k)$. Every vertex is associated to a binary string of length $k$. Hence, it will be linked to all (and only) the vertices associated to string obtained by changing exactly one digit (for example, the vertex $000\dots 0$ will be adjacent to $10\dots 0$, $010\dots 0$ etc.) It means that every vertex is linked to exactly $k$ other vertices. So $d(Q_k)=k$.
Notice that the number of binary strings of length $k$ is $2^k$, so $|Q_k|=2^k$.
Since we determined the number of nodes and the number of edges per node, it is easy to find $e(Q_k)=d(Q_k)|Q_k|/2=k2^{k-1}$.
In order to show that it is bipartite, just notice that a vertex with an odd number of $0$s con only be adjacent to a vertex with n even number of $0$s, since you require that they differ in exactly one place. Hence, the sets of vertices $A:=\{x:x$ has an even number of $0$s$\}$ and $B:=\{x:x$ has an odd number of $0$s$\}$ describe a bipartition of the graph.