# The Graph Menagerie: Abstract Algebra and the Mad Veterinarian

• Published in 2011
In the collections
This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed "Mad Vet" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.

### BibTeX entry

@article{TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian,
title = {The Graph Menagerie: Abstract Algebra and the Mad Veterinarian},
author = {Gene Abrams and Jessica K. Sklar},
}