We can divide the nodes of the graph into two disjoint sets A and B, such that there is no edge between the vertices within a set.
Say bit string length is 3, then total vertices = 8.
Include vertex 000 into set A and include vertex 001 into set B,
Now include vertex V into set A, if the Hamming distance between V and 000 is even.
Same way include vertex V into set B, if the Hamming distance between V and 001 is even.
Now Set A = {000,011,110,101}
Set B = {001,111,100,010}
It makes sure there is no edge between the vertices within a set(because every pair of vertices in a set has an even hamming distance)
So every Bipartite graph can be colored with 2 colors.
And the shortest distance between every pair of vertices is the hamming distance between the vertices.
And the largest of such distance i.e diameter is between 000 and 111.
So the diameter of the graph is N.
It’s indeed an N-Cube graph containing 2^N vertices.
More about the N-Cube graph :
N-Cube graph is bipartite and can be colored with 2 colors.
The degree of each vertex is N(i.e total vertices having the hamming distance of 1 with the respective vertex)
The diameter of an N-Cube graph is N.
PS : You can relate the N-Cube graph with the concept of Boolean Algebra in Group theory.