An Explicit Infinite Family of $$\mathbb{M}$$-Vertex Graphs with Maximum Degree K and Diameter $$\left[ {1 + o\left( 1 \right)} \right]{\log _{K - 1}\mathbb{M}}$$ for Each K − 1 a Prime Power

Abstract

We present the solution of a long-standing open question by giving an explicit construction of an infinite family of M-vertex cubic graphs that have diameter [1 + o(1)] log 2 M. Then, for every K in the form K = p s + 1, where p can be any prime [including 2] and s any positive integer, we extend the techniques to construct an infinite family of K-regular graphs on M vertices with diameter [1 + o(1)] log K−1 M.