Thread: DIAMETER OF HAMMING GRAPH H(d, 3)

Hello!

My question is very simple. I know that the diameter of Hamming graph H(d, 3) is d.
I just want to know how to prove this?

Thanks for answers ...

Sketch: Let ( ) and ( ) vertices in the graph. Then ( ) is adjacent to ( ). Continuing we have that a path of length d exists from ( ) to ( ). Thus the minimum eccentricity of a vertex is d. But note that we can do this for any pair so the max eccentricity is d as well. It follows that the diameter is d.

did you mean in second sentence that (a1, a2, ..., ad) is adjacent to (b1, b2, ..., bd)?

No, what is written is correct.
Because we are in a Hamming Graph vertices are adjacent if and only if they differ by a single term. So what I am doing is constructing a path using vertices that are known to be adjacent. So constructing the path I am describing actually has 'd' steps, but I only explicitly showed one. (This is the main reason the eccentricity is 'd')