Minimum tree-stretch of Hamming graphs and higher-dimensional grids

The minimum stretch spaning tree problem for a grah G is to find a spaning tree T of G such as that the maximum distance in T between two adjacent vertices is minimized. The minimum value of this optimization problem gives rise to a grpah invariant {\sigma} T(G) called the tree stretch of G. The problem has been studied in the algorithmic aspects, such as NP-hardness and fixed-parameter solvability. This paper presents the exact values {\sigma} T(G) of hamming graphs Kn1 * Kn2 * ... * Knd and the higer-dimensional grids Pn1 * Pn2 * ... * Pnd.