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A combinatorial introduction to Adinkras

Robert W. Donley, S. James Gates, Tristan Hübsch, Rishi Nath

Abstract

We survey the combinatorics of the Adinkra, a graphical device for solving differential equations in supersymmetry. These graphs represent an exceptional class of 1-factorizations with further augmentations. As a new feature, we characterize Adinkras using Latin rectangles.

A combinatorial introduction to Adinkras

Abstract

We survey the combinatorics of the Adinkra, a graphical device for solving differential equations in supersymmetry. These graphs represent an exceptional class of 1-factorizations with further augmentations. As a new feature, we characterize Adinkras using Latin rectangles.

Paper Structure

This paper contains 9 sections, 13 theorems, 8 equations, 11 figures.

Table of Contents

  1. Introduction
  2. 1-factorizations and Regular Edge Colorings
  3. The Quadrilateral Property
  4. Adjacency Lists and Latin Rectangles
  5. Adjacency Matrices and Semi-magic Squares
  6. The Hypercube Graphs $Q_N$ and their Quotients
  7. Totally Odd Dashings
  8. Partial Orderings
  9. Motivation from Supersymmetry

Key Result

corollary 1

If a graph $G$ is bipartite and $N$-regular, then $G$ admits a 1-factorization.

Figures (11)

  • Figure 1: A regular edge coloring for $K_4$ with its associated perfect matchings
  • Figure 2: A regular edge coloring for $K_{4, 4}$
  • Figure 3: The quadrilateral property for $K_4$
  • Figure 4: A regular edge coloring for $Q_3$
  • Figure 5: Adjacency lists $L(c)$ for $K_4$ and $Q_3$
  • ...and 6 more figures

Theorems & Definitions (42)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • corollary 1
  • proposition 1
  • definition 5
  • definition 6
  • remark 1
  • definition 7
  • ...and 32 more