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

Abstract

Let n ≥ k ≥ 2 be two integers and S a subset of {0, 1, . . . , k − 1}. The graph J S (n, k) has as vertices the k-subsets of the n-set [n] = {1, . . . , n} and two k-subsets A and B are adjacent if |A ∩ B| ∈ S. In this paper, we use Godsil-McKay switching to prove that for m ≥ 0, k ≥ 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 ≥ 2, n ≥ 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 ≤ 5.

On the Diameter and Zero Forcing Number of Some Graph Classes in the Johnson, Grassmann and Hamming Association Scheme

“…, t} for some integer t ≤ k − 1. This generalization also appears in [7], where the coloring number of generalized Kneser graphs is studied, and [17], which shows that some of these graphs are not determined by their spectrum. As usual for Kneser graphs, we require n ≥ 2k + 1.…”

New strongly regular graphs from finite geometries via switching

“…Two graphs Γ and Γ are cospectral if their adjacency matrices A and A are cospectral, that is there exists an orthogonal matrix Q with Q T AQ = A. In 1982 Godsil and McKay described a possible choice for Q which has an easy combinatorial description [10] -nowadays known as Godsil-McKay switching -and proved to be very useful in constructing cospectral graphs [2,7,8,9,11,20,21]. Godsil-McKay switching can be described as follows.…”