Graphs cospectral with Kneser graphs

Abstract

We construct graphs that are cospectral but nonisomorphic with Kneser graphs K(n, k), when n = 3k − 1, k > 2 and for infinitely many other pairs (n, k). We also prove that for 3 ≤ k ≤ n − 3 the Modulo-2 Kneser graph K2(n, k) is not determined by the spectrum.

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

“…Focusing mainly on Kneser graphs, we were able to eliminate the possibility of switching sets of size 8 in K(9, 3), K(10, 3), K(11, 3), K(12, 3) and K(10, 4) as well as switching sets of size 10 in K(9, 3) and K(10, 3). Our computations extend the computations of Haemers and Ramezani [8] which did not find any switching sets of size 4 or 6 in the Kneser graphs K(9, 3) nor K (10,3). At present time, these are smallest graphs in the Johnson scheme whose spectral characterization is not known.…”

“…The other graph we found switching sets for is J {2} (8,4), however none of these switching sets produced nonisomorphic cospectral mates. However, it is interesting to note that the switching sets we found were of size 4.…”

“…Taking m = 0 in Theorem 6, we obtain the switching sets found for Kneser graphs K(3k − 1, k) in [8].…”

“…In [8] it was shown that taking C to be the set of vertices containing [k − 2] as a subset is a switching set for the graphs K(n, k) satisfying 2n = 6k − 3 + √ 8k 2 + 1. It is reasonable to try to generalize this in a way similar to what we have done for K(3k − 1, k).…”

Cospectral constructions and spectral properties of variations of the distance matrix

“…This cospectral construction method is the most well known because of its prevalence in explaining cospectrality, as discussed by Haemers and Spence [30]. Further, it has been used in special cases to find cospectral graphs with certain graph structure: [5,13,15,19,22,24,27,28,50,51].…”