The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

Brouwer, Andries E.; Cioabă, Sebastian M.; Ihringer, Ferdinand; McGinnis, Matt

doi:10.1016/j.jctb.2018.04.005

Cited by 27 publications

(30 citation statements)

References 29 publications

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“…In the following table, we list this divisibility condition for the association schemes we study in the following sections (see the corresponding sections for the notation). We refer to [4, §9] and [3] for the eigenvalues.…”

Section: Association Schemesmentioning

confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

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Section: Association Schemesmentioning

confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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“…1. The proof of the second largest eigenvalue in absolute value of J(n, k, t) was recently confirmed in [BCIM18] to be as follows:…”

Section: A Missing Proofsmentioning

confidence: 89%

Toward a General Direct Product Testing Theorem

Goldenberg

Karthik

2019

ACM Trans. Comput. Theory

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The Direct Product encoding of a string a ∈ {0, 1} n on an underlying domain V ⊆ ( [n] k ), is a function DP V (a) which gets as input a set S ∈ V and outputs a restricted to S. In the Direct Product Testing Problem, we are given a function F : V → {0, 1} k , and our goal is to test whether F is close to a direct product encoding, i.e., whether there exists some a ∈ {0, 1} n such that on most sets S, we have F(S) = DP V (a)(S). A natural test is as follows: select a pair (S, S ′ ) ∈ V according to some underlying distribution over V × V, query F on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph.The testability of direct products was studied over various domains and test graphs: Dinur and Steurer (CCC'14) analyzed it when V equals the k-th slice of the Boolean hypercube and the test graph is a member of the Johnson graph family. Dinur and Kaufman (FOCS'17) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case V = O k (n).In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem?Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every k and n we provide a direct product domain V ⊆ ( n k ) of size n, called the Sliding Window domain for which we prove direct product testability.

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“…If G is a simple graph of order n, not necessarily regular, results of Lovász [31,Theorems 6,10 ] imply that…”

Section: Each Pair Of Distinct Vertices In U Is In At Most One Cliquementioning

confidence: 99%

“…Since the spectrum of a connected graph is symmetric if and only if the graph is bipartite, it is natural to think of λ(G) as a measure of how bipartite G is. It is therefore not surprising that the smallest eigenvalue has close connections to the max-cut [5,10,24,26]. There are several methods to obtain upper bounds for λ(G).…”

Section: Introductionmentioning

confidence: 99%

Some observations on the smallest adjacency eigenvalue of a graph

Cioabă¹,

Elzinga²,

Gregory³

2020

Discuss. Math. Graph Theory

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In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.A 3 = rA + s(J − I) + tI for some nonnegative integers r, s, t depending on n, k, a, c. ThusThus, z is at least as large as the minimum root of the cubic x 3 − rx + s − t. Two of the roots are θ and τ , inherited from (5), and the other is necessarily −(θ + τ ) = c − a since the coefficient of x 2 is 0. Thus, λ(G) = z ≥ min{c − a, τ }. Therefore, taking G (3) in Theorem 2.1 gives λ(G) = τ when G is a strongly regular graph such that τ ≤ c − a. In particular, λ(G) = τ when a ≤ c, a condition that must be satisfied by at least one of a strongly regular graph and its complement.

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The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

Cited by 27 publications

References 29 publications

Boolean degree 1 functions on some classical association schemes

Boolean degree 1 functions on some classical association schemes

Toward a General Direct Product Testing Theorem

Some observations on the smallest adjacency eigenvalue of a graph

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