Extremal matrices for the Bruhat-graph order

UID/MAT/00297/2019 UID/MAT/04721/2019 We consider the class (Formula presented.) of symmetric (Formula presented.) -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 (Formula presented.) the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on (Formula presented.) and study the minimal and maximal matrices when k = 1, 2 or 3. publishersversion published