Some Metrical Properties of Lattice Graphs of Finite Groups

This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . We show that the lattice graph of Z p 1 × p 2 × ⋯ × p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.