Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Switching graphs and Hadamard matrices

Aida Abiad, Louka Peters

Abstract

Local operations of combinatorial structures (graphs, Hadamard matrices, codes, designs) that maintain the basic parameters unaltered, have been widely used in the literature under the name of switching. We show an equivalence between two switching methods to construct inequivalent Hadamard matrices, which were proposed by Orrick [SIAM Journal on Discrete Mathematics, 2008], and the switching method for constructing cospectral graphs which was introduced by Godsil and McKay [Aequationes Mathematicae, 1982].

Switching graphs and Hadamard matrices

Abstract

Local operations of combinatorial structures (graphs, Hadamard matrices, codes, designs) that maintain the basic parameters unaltered, have been widely used in the literature under the name of switching. We show an equivalence between two switching methods to construct inequivalent Hadamard matrices, which were proposed by Orrick [SIAM Journal on Discrete Mathematics, 2008], and the switching method for constructing cospectral graphs which was introduced by Godsil and McKay [Aequationes Mathematicae, 1982].

Paper Structure

This paper contains 19 sections, 9 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Switching to construct cospectral graphs
  4. Switching to construct Hadamard matrices
  5. Switching a closed quadruple
  6. Switching a Hall set
  7. Graphs from Hadamard matrices: Hadamard graphs
  8. New equivalences
  9. Switching a closed quadruple and GM-switching
  10. Switching a Hall set and GM-switching
  11. Non-equivalence of Hadamard matrices constructed using Orrick's switchings
  12. Switching a closed quadruple
  13. Switching a Hall set
  14. Concluding remarks
  15. Appendix
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $G$ be a graph and consider $\{C_1,\dots,C_t,D\}$ to be a partition of its vertices such that, for all $i,j\in\{1,\dots,t\}$: For all $i\in\{1,\dots,t\}$ and every $v\in D$ that has exactly $\frac{1}{2}|C_i|$ neighbours in $C_i$, swap the adjacencies between $v$ and $C_i$. The resulting graph is cospectral with $G$.

Figures (2)

  • Figure 1: Hadamard graphs corresponding to $H_1, H_2$ and $H_3$, respectively.
  • Figure 2: Subgraph corresponding to the switched field before and after switching a closed quadruple.

Theorems & Definitions (20)

  • Theorem 2.1: GM-switching Godsil1982
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 10 more