Switching methods of level 2 for the construction of cospectral graphs

TL;DR

This paper introduces new switching methods for constructing cospectral graphs, classifies irreducible methods, and extends previous results related to regular orthogonal matrices of level 2.

Contribution

It presents two novel switching methods, offers reformulations of existing operations, and classifies irreducible methods up to size 12.

Findings

Two new switching methods are introduced.

A classification of irreducible switching methods is provided.

Results extend previous classifications up to switching sets of size 12.

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.