A geometrical characterization of strongly regular graphs

Nozaki, Hirōshi; Shinohara, M.

doi:10.1016/j.laa.2012.05.040

Cited by 10 publications

(11 citation statements)

References 29 publications

(34 reference statements)

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“…Nozaki and Shinohara [22] also give necessary and sufficient conditions of a Euclidean representation of G to be spherical. However, their conditions are more bulky.…”

Section: Introductionmentioning

confidence: 99%

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Graphs and spherical two-distance sets

Musin

2019

European Journal of Combinatorics

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Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical representation numbers of G and give exact formulas for these numbers using multiplicities of polynomials that are defined by the Caley-Menger determinant. One of the main results of the paper are explicit formulas for the representation numbers of the join of graphs which are obtained from W. Kuperberg's type theorem for two-distance sets.Throughout this paper we will consider only simple graphs, R d will denote the ddimensional Euclidean space, S n will denote the n-dimensional unit sphere in R n+1 , and dist(x, y) := ||x − y|| will denote the Euclidean distance in R d . For a set X ⊂ R d we shall denote the affine hull (or affine span) by aff(X), rank(X) := dim aff(X) and conv(X) will denote the convex hull of X. We will denote the cardinality of a finite set X by |X| .

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“…Nozaki and Shinohara [22] also give necessary and sufficient conditions of a Euclidean representation of G to be spherical. However, their conditions are more bulky.…”

Section: Introductionmentioning

confidence: 99%

“…However, their conditions are more bulky. Namely, they used Roy's theorem (see [22,Theorem 2.4]) and they showed that among five types of conditions only three of them yields sphericity [22,Theorem 3.7].…”

Section: Introductionmentioning

confidence: 99%

Graphs and spherical two-distance sets

Musin

2019

European Journal of Combinatorics

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“…In this paper, we extend and simplify the results of Roy [18] and Nozaki and Shinohara [17] by deriving exact simple formulas for dim E (G) and dim S (G) in terms of the multiplicities of the smallest and the largest eigenvalues of the (n − 1) × (n − 1) matrix V T AV , where A is the adjacency matrix of G and V , defined in (3), is the matrix whose columns form an orthonormal basis for the orthogonal complement of the vector of all 1's. This is made possible by using projected Gram matrices for representing n-point configurations.…”

mentioning

confidence: 52%

On representations of graphs as two-distance sets

Alfakih

2020

Discrete Mathematics

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Let α = β be two positive scalars. A Euclidean representation of a simple graph G in R r is a mapping of the nodes of G into points in R r such that the squared Euclidean distance between any two points is α if the corresponding nodes are adjacent and β otherwise. A Euclidean representation is spherical if the points lie on an (r − 1)-sphere, and is J-spherical if this sphere has radius 1 and α = 2 < β. Let dim E (G), dim S (G) and dim J (G) denote, respectively, the smallest dimension r for which G admits a Euclidean, spherical and J-spherical representation.In this paper, we extend and simplify the results of Roy [18] and Nozaki and Shinohara [17] by deriving exact simple formulas for dim E (G) and dim S (G) in terms of the eigenvalues of V T AV , where A is the adjacency matrix of G and V is the matrix whose columns form an orthonormal basis for the orthogonal complement of the vector of all 1's. *We also extend and simplify the results of Musin [16] by deriving explicit formulas for determining the J-spherical representation of G and for determining dim J (G) in terms of the largest eigenvalue ofĀ, the adjacency matrix of the complement graphḠ. As a by-product, we obtain several other related results and in particular we answer a question raised by Musin in [16].

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“…Observe that a related result was proved in [20]. Namely, Theorem 4.7 in that paper states (in our terms) that a spherical set S ⊂ R n is a 2-design if and only if G · 1 = 0 and G 2 = x∈S x 2 n G.…”

Section: Two-distance Funtfs and Strongly Regular Graphsmentioning

confidence: 87%

Finite two-distance tight frames

Barg

Glazyrin

Okoudjou

et al. 2015

Linear Algebra and its Applications

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A finite collection of unit vectors S ⊂ R n is called a spherical two-distance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We prove that if a = −b, then a two-distance set that forms a tight frame for R n is a spherical embedding of a strongly regular graph. We also describe all two-distance tight frames obtained from a given graph. Together with an earlier work by S. Waldron (2009) [22] on the equiangular case, this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all twodistance 2-designs.

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A geometrical characterization of strongly regular graphs

Cited by 10 publications

References 29 publications

Graphs and spherical two-distance sets

Graphs and spherical two-distance sets

On representations of graphs as two-distance sets

Finite two-distance tight frames

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