AI 中文总结
本文研究域上方阵矩阵空间的奇异差图Γ，证明其为直径2的连通正则图，在有限域上推导顶点度数公式与欧拉图判定条件，确定独立数和团数并构造对应结构，还给出控制数上界。
AI 中文摘要
域上方阵矩阵构成的向量空间的奇异差图记为Γ，该图的顶点集为该向量空间的所有元素，两个不同顶点相邻当且仅当对应矩阵的差是奇异的。本文研究Γ的基本图论性质，包括连通性、直径、正则性、欧拉性、独立数、团数和控制数。证明Γ是连通正则图，直径为2；在有限域上得到每个顶点度数的显式公式，精确刻画Γ为欧拉图的条件；确定独立数和团数，利用不可约多项式的友矩阵构造达到这些值的显式结构；还构造显式控制集，给出控制数的上界。
英文摘要
The singular difference graph, denoted by $Γ$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $Γ$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $Γ$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $Γ$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.
Comments13 pages
