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 ($\displaystyle a_1, a_2, ..., a_d)$) and ($\displaystyle b_1, b_2, ..., b_d)$) vertices in the graph. Then ($\displaystyle a_1, a_2, ..., a_d)$) is adjacent to ($\displaystyle b_1, a_2, ..., a_d)$). Continuing we have that a path of length d exists from ($\displaystyle a_1, a_2, ..., a_d)$) to ($\displaystyle b_1, b_2, ..., b_d)$). 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.
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')