We consider the class ${\mathcal{A}}_{\mathrm{s}\mathrm{y}\mathrm{m}}^0(n,k)$ of symmetric $(0,1)$-matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for $k\ge 3$ the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on ${\mathcal{A}}_{\mathrm{s}\mathrm{y}\mathrm{m}}^0(n,k)$ and study the minimal and maximal matrices when k = 1, 2 or 3.

中文翻译：

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Bruhat图阶的极值矩阵

我们考虑上课 ${\mathcal{一种}}_{\mathrm{s}\mathrm{ÿ}\mathrm{米}}^0（ñ，ķ）$ 对称的 $（0，\mathrm{1个}）$具有零迹线和恒定行总和k的-矩阵，可以用k-正则无向图的邻接矩阵的类别来标识。在先前的论文中，此类中引入了两个部分阶，即Bruhat和Bruhat-graph阶。实际上，当k  = 1或k  = 2时，表明两个阶数是重合的，而对于$ķ\ge 3$这两个顺序是不同的。在本文中，我们给出了这些阶上的最小和最大矩阵的一般性质${\mathcal{一种}}_{\mathrm{s}\mathrm{ÿ}\mathrm{米}}^0（ñ，ķ）$并研究k  = 1、2或3时的最小和最大矩阵。