Graphs Cospectral with Kneser Graphs

Haemers, Willem H.; Ramezani, Farzaneh

doi:10.2139/ssrn.1479567

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…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.…”

Section: Computational Resultssupporting

confidence: 75%

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

Section: Kneser and Johnson Graphsmentioning

confidence: 99%

“…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).…”

Section: Vertices Containing [K − 2] As a Possible Switching Setmentioning

confidence: 99%

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

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Haemers

Johnston

et al. 2018

Linear Algebra and its Applications

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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.

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Section: Computational Resultssupporting

confidence: 75%

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

Section: Kneser and Johnson Graphsmentioning

confidence: 99%

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

Cioabă

Haemers

Johnston

et al. 2018

Linear Algebra and its Applications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

New strongly regular graphs from finite geometries via switching

Ihringer

Munemasa

2019

Linear Algebra and its Applications

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We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U (n, 2), O(n, 3), O(n, 5), O + (n, 3), and O − (n, 3) are not determined by its parameters for n ≥ 6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.

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“…For a survey of results on this area we refer the reader to [4,5]. Some families of regular graphs, specially distance regular graphs have attracted more attention, see for instance [1,6,8,9,11].…”

Section: Introductionmentioning

confidence: 99%