AI 中文总结
本文研究有限域Fp上的4阶单位上三角群UT(4; p)的交换图，明确其交换关系，证明约化交换图连通且直径为3，还确定了其最大团大小、色数等图论性质。
AI 中文摘要
设G = UT(4; p)为有限域Fp上所有4×4单位上三角矩阵构成的群，其中p为素数。利用G元素的六参数形式，我们明确描述其交换关系并以此分析图的结构。我们证明约化交换图是连通的，直径为3。此外，通过将图分解为陪集、层和方向部分，我们确定了其最大团的大小、色数、独立数及完美性等性质。
英文摘要
Let G = UT(4; p) be the group of all 4 ? 4 unitriangular matrices over the fi?nite fi?eld Fp, where p is a prime. Using the six-parameter form of the elements of G, we describe the commutativity relation explicitly and use it to analyse the struc- ture of the graph. We prove that the reduced commuting graph is connected and has diameter 3. We also determine the size of its maximal cliques, chromatic number, independence number, per- fectness, etc. by decomposing the graph into cosets, layers, and direction parts.
