# The Graph Menagerie: Abstract Algebra and the Mad Veterinarian

- Published in 2011
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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.

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### BibTeX entry

@article{TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian, title = {The Graph Menagerie: Abstract Algebra and the Mad Veterinarian}, author = {Gene Abrams and Jessica K. Sklar}, url = {https://www.maa.org/programs/maa-awards/writing-awards/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian https://www.maa.org/sites/default/files/pdf/upload{\_}library/22/Allendoerfer/Abrams2011.pdf}, urldate = {2019-11-06}, year = 2011, abstract = {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.}, comment = {}, collections = {Animals,Attention-grabbing titles,Easily explained,Puzzles} }