## Abstract

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 language | English |
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Pages (from-to) | 49-64 |

Number of pages | 16 |

Journal | Discrete Applied Mathematics |

Volume | 258 |

DOIs | |

Publication status | Published - 15 Apr 2019 |

## Keywords

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