# Cospectral mates for the union of some classes in the Johnson association scheme

## TL;DR

This paper demonstrates that certain Johnson scheme graphs are not uniquely determined by their spectra using Godsil-McKay switching, and reports computational searches for such switching sets.

## Contribution

It introduces new non-isomorphic graphs with identical spectra within Johnson schemes using Godsil-McKay switching, expanding understanding of spectral graph characterization.

## Key findings

- Graphs $J_S(3k-2m-1,k)$ are not spectrum-determined for specified parameters.
- Graphs $J_{S}(n,2m+1)$ are not spectrum-determined under certain conditions.
- Computational searches identified potential switching sets in Johnson scheme unions for small $k$.

## Abstract

Let $n\geq k\geq 2$ be two integers and $S$ a subset of $\{0,1,\dots,k-1\}$. The graph $J_{S}(n,k)$ has as vertices the $k$-subsets of the $n$-set $[n]=\{1,\dots,n\}$ and two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|\in S$. In this paper, we use Godsil-McKay switching to prove that for $m\geq 0$, $k\geq \max(m+2,3)$ and $S = \{0, 1, ..., m\}$, the graphs $J_S(3k-2m-1,k)$ are not determined by spectrum and for $m\geq 2$, $n\geq 4m+2$ and $S = \{0,1,...,m\}$ the graphs $J_{S}(n,2m+1)$ are not determined by spectrum. We also report some computational searches for Godsil-McKay switching sets in the union of classes in the Johnson scheme for $k\leq 5$.