# DIAMETER OF HAMMING GRAPH H(d, 3)

• May 14th 2012, 01:35 PM
lebdim
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?

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