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Switching methods of level 2 for the construction of cospectral graphs

Aida Abiad, Nils van de Berg, Robin Simoens

Abstract

A switching method is a graph operation that results in cospectral graphs (graphs with the same spectrum). Work by Wang and Xu [Discrete Math. 310 (2010)] suggests that most cospectral graphs with cospectral complements can be constructed using regular orthogonal matrices of level 2, which has relevance for Haemers' conjecture. We present two new switching methods and several combinatorial and geometrical reformulations of existing switching operations of level 2. We also introduce the concept of reducibility and use it to classify all irreducible switching methods that correspond to a conjugation with a regular orthogonal matrix of level 2 with one nontrivial indecomposable block, up to switching sets of size 12, extending previous results.

Switching methods of level 2 for the construction of cospectral graphs

Abstract

A switching method is a graph operation that results in cospectral graphs (graphs with the same spectrum). Work by Wang and Xu [Discrete Math. 310 (2010)] suggests that most cospectral graphs with cospectral complements can be constructed using regular orthogonal matrices of level 2, which has relevance for Haemers' conjecture. We present two new switching methods and several combinatorial and geometrical reformulations of existing switching operations of level 2. We also introduce the concept of reducibility and use it to classify all irreducible switching methods that correspond to a conjugation with a regular orthogonal matrix of level 2 with one nontrivial indecomposable block, up to switching sets of size 12, extending previous results.

Paper Structure

This paper contains 20 sections, 25 theorems, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Godsil-McKay switching
  4. Wang-Qiu-Hu switching
  5. Switching methods of level 2
  6. The approach
  7. Abiad-Haemers switching
  8. Mao-Wang-Liu-Qiu switching or Sun graph switching
  9. Two new switching methods of level 2
  10. Reducibility
  11. Classification of irreducible switching methods of level 2 with one nontrivial indecomposable block, up to size 12
  12. Irreducible switching methods from the infinite family
  13. Six vertex AH-switching
  14. Eight vertex AH-switching (from the infinite family)
  15. Ten vertex switching
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Gamma$ be a graph and let $\{C_1,\dots,C_t,D\}$ 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 $\mathbb{R}$-cospectral with $\Gamma$.

Figures (7)

  • Figure 1: Cospectral mates, obtained by Six vertex AH-switching.
  • Figure 2: Cospectral mates, obtained by Fano switching.
  • Figure 3: Cospectral mates, obtained by Cube switching.
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (50)

  • Theorem 1: GM-switching GMswitching
  • Theorem 2: WQH-switching WQHswitching
  • Theorem 3: weighing
  • Definition 4
  • Theorem 5: Sun graph switching MAO2023
  • Lemma 6: phdaida
  • Theorem 7
  • proof
  • Theorem 8
  • proof
  • ...and 40 more