Tree spanners, Cayley graphs, and diametrically uniform graphs

In line with symmetrical graphs such as Cayley graphs and vertex transitive graphs, we introduce a new class of symmetrical graphs called diametrically uniform graphs. The class of diametrically uniform graphs includes vertex transitive graphs and hence Cayley graphs. A tree t-spanner of graph G is a spanning tree T in which the distance between every pair of vertices is at most t times their distance in G. The minimum tree spanner problem of a graph G is to find a tree t-spanner with t as small as possible. In this paper, the minimum tree spanner problem is exhaustively studied for diametrically uniform graphs, which also include 3-regular mesh of trees and generalized Petersen graphs.