Extremal matrices for the Bruhat-graph order

Abstract

We consider the class A 0 sym (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 Bruhatgraph 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 ≥ 3 the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on A 0 sym (n, k) and study the minimal and maximal matrices when k = 1, 2 or 3.

The Bruhat order on classes of isotopic Latin squares

“…The Bruhat order on (0, 1)-matrices is receiving the attention of many researchers, [4,6,10,11,12,13,15,16,17,18,19]. In the recent years several authors have taken Brualdi and Hwang's ideas, and extended the Bruhat order to other classes of matrices than (0, 1)-matrices: the Bruhat order has been studied on the class of tournament matrices with a given score vector, [8], on the class of alternating sign matrices, [9], and on the class of doubly stochastic matrices, [7].…”

On the little secondary Bruhat order

“…Doing this they gave rise to two distinct partial order relations on A(R, S). In the last 16 years, many research have focused on several topics of these two partial order relations: conjectures [15], minimal elements [2,4,16], coincidence [12], chains and antichains [9,10,20,21], restrictions of the Bruhat order on subclasses of A(R, S) [8,11], or extensions of one of these orders to other classes of matrices distinct of A(R, S) [5,6,7,13,14].…”