On the Bruhat order of labeled graphs

Richard A. Brualdi, Rosário Fernandes, Susana Furtado

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)
25 Downloads (Pure)


We investigate two Bruhat (partial) orders on graphs with vertices labeled 1,2,…,n and with a specified degree sequence R, equivalently, symmetric (0,1)-matrices with zero trace and a specified row sum vector R (adjacency matrices of such graphs). One is motivated by the classical Bruhat order on permutations while the other one, more restrictive, is defined by a switch of a pair of disjoint edges. In the Bruhat order, one seeks to concentrate the edges of a graph with a given degree sequence among the vertices with smallest labels, thereby producing a minimal graph in this order. We begin with a discussion of graphs whose isomorphism class does not change under a switch. Then we are interested in when the two Bruhat orders are identical. For labeled graphs of regular degree k, we show that the two orders are identical for k≤2 but not for k=3.

Original languageEnglish
Pages (from-to)49-64
Number of pages16
JournalDiscrete Applied Mathematics
Publication statusPublished - 15 Apr 2019


  • Adjacency matrix
  • Bruhat order
  • Degree sequence
  • Labeled graph
  • Switching
  • Symmetric matrix


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