We present the solution of a long-standing open question by giving an explicit construction of an infinite family of \(\mathbb{M}\)-vertex cubic graphs that have diameter \(\left[ {1 + o\left( 1 \right)} \right]{\log _2}\mathbb{M}\). 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 \(\mathbb{M}\) vertices with diameter \(\left[ {1 + o\left( 1 \right)} \right]{\log _{K - 1}\mathbb{M}}\).

中文翻译：

---

具有最大度数K和直径$$ \ left [{1 + o \ left（1 \ right）} \ right] {\ log _ {K -1} \ mathbb {M}} $$ [1 + o（1）] log K − 1 M，每K − 1 a素数

我们通过给无限家族的明确结构呈现长期悬而未决的问题的解决方案\（\ mathbb {M} \） -点立方图表，具有直径\（\左[{1 + O \左（1 \ right）} \ right] {\ log _2} \ mathbb {M} \）。然后，对于每ķ形式ķ = p^小号+ 1，其中p可以是任何素[包括2]和š任意正整数，我们扩展的技术来构建的无限族ķ上-regular图表\（\ mathbb直径为\（\ left [{1 + o \ left（1 \ right）} \ right] {\ log _ {K-1} \ mathbb {M}} \ }的{M} \）个顶点。