# [Nauty] Matrix isomorphism

Gordon Royle gordon.royle at uwa.edu.au
Thu Aug 18 11:15:11 AEST 2016

```Any such matrix can be viewed as the bipartite adjacency matrix of a graph - i.e. the rows are the vertices of one colour, and the columns are the vertices of the second colour, and conversely any bipartite graph yields such a matrix (or two matrices if you care about which is rows, which is columns).

Isomorphisms between such graphs that preserve the colours correspond to permuting the rows and the columns of the matrix, but not exchanging them. Isomorphisms that are permitted to swap the colours correspond to matrix isomorphisms swapping the rows and columns.

On 18 Aug 2016, at 8:34 am, Wayne Kelly <w.kelly at qut.edu.au<mailto:w.kelly at qut.edu.au>> wrote:

Hi,

I'm trying to enumerate binary (0-1) matrices of a given size that satisfying a specific matrix property (related to finite projection planes), but want to exclude isomorphic matrices, i.e. I don't care about the row or column order of the matrix.

Can nauty help me with this problem? I realize that nauty is designed to work with Graphs rather than Matrices, but thought the same kinds of techniques may be useful.

Cheers, Wayne.

Dr Wayne Kelly | Senior Lecturer
Science and Engineering Faculty | Queensland University of Technology

S Block, Level 10, Room S-1011 (enter via S-1013), Gardens Point Campus
ph 3138 9336 | email w.kelly at qut.edu.au<mailto:w.kelly at qut.edu.au>
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Professor Gordon Royle
School of Mathematics and Statistics
University of Western Australia
Gordon.Royle at uwa.edu.au<mailto:Gordon.Royle at uwa.edu.au>

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