Distance-Regular Graphs

Brouwer, Andries E.; Cohen, Arjeh M.; Neumaier, Arnold

doi:10.1007/978-3-642-74341-2

Cited by 1,563 publications

(2,425 citation statements)

References 281 publications

(734 reference statements)

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“…• The following SRGs are not observable: 20,1,6)), Higman-Sims (SRG(100,22,0,6)), M22 (SRG(77,16,0,4)), Hoffman-Singleton (SRG(50,7,0,1)), Sims-Gewirtz (SRG(56,10,0,2)). (see Godsil and Royle (2001) and Brouwer et al (1989) for more descriptions of these graphs).…”

Section: On Observability Of Discrete-time Networked Systemsmentioning

confidence: 99%

Decentralized Laplacian Eigenvalues Estimation and Collaborative Network Topology Identification

Kibangou

Commault

2012

IFAC Proceedings Volumes

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In this paper we first study observability conditions on networks. Based on spectral properties of graphs, we state new sufficient or necessary conditions for observability. These conditions are based on properties of the Khatri-Rao product of matrices. Then we consider the problem of estimating the eigenvalues of the Laplacian matrix associated with the graph modeling the interconnections between the nodes of a given network. Eventually, we extend the study to the identification of the network topology by estimating both eigenvalues and eigenvectors of the network matrix. In addition, we show how computing, in finite-time, some linear functionals of the state initial condition, including average consensus. Specifically, based on properties of the observability matrix, we show that Laplacian eigenvalues can be recovered by solving a local eigenvalue decomposition on an appropriately constructed matrix of observed data. Unlike FFT based methods recently proposed in the literature, in the approach considered herein, we are also able to estimate the multiplicities of the eigenvalues. Then, for identifying the network topology, the eigenvectors are estimated by means of a consensus-based least squares method.

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Section: On Observability Of Discrete-time Networked Systemsmentioning

confidence: 99%

Decentralized Laplacian Eigenvalues Estimation and Collaborative Network Topology Identification

Kibangou

Commault

2012

IFAC Proceedings Volumes

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“…-take Q 2k−1 and identify every two diametrical vertices of it; therefore they are also called folded (2k − 1)-cubes, see [3]; -take Q 2k−2 and add to it 2 2k−3 "diagonals" (i.e. new edges, joining diametrical vertices of Q 2k−2 ).…”

Section: Examplesmentioning

confidence: 99%

Spherical and Clockwise Spherical Graphs

Berrachedi

Havel

Mulder

2003

Czechoslovak Mathematical Journal

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“…This program has been implemented in the case of strongly regular graphs with the eigenvalue 2 (see [3]). An antipodal distance regular graph Γ of diameter 3 has (see [1]) the intersection array {k, μ(r -1), 1; 1, μ, k} and possesses v = r(k + 1) vertices and the spectrum k b …”

mentioning

confidence: 99%

“…An antipodal distance regular graph Γ of diameter 3 has (see [1]) the intersection array {k, μ(r -1), 1; 1, μ, k} and possesses v = r(k + 1) vertices and the spectrum k b…”

mentioning

confidence: 99%

On distance-regular graphs in which neighborhoods of vertices are strongly regular

2013

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We consider undirected graphs without loops or multiple edges. Given a vertex a in a graph Γ, let Γ i (a) denote the i neighborhood of a, i.e., the subgraph induced by Γ on the set of all its vertices that are a dis tance of i away from a. Let [a] = Γ 1 (a) and a ⊥ = {a} ∪ [a].A strongly regular graph with parameters (v, k, λ, μ) and with integer eigenvalues k, r, and s, where s < 0, is called a Smith graph if A nonbipartite graph for which the fundamental bound becomes an equality is called tight. The neigh borhood of any vertex in a tight graph is a strongly regular graph with eigenvalues a 1 , bIt is well known that a tight graph of diameter 3 is a Taylor graph (see, e.g., [2, Theorem 3.2]). In this case, the neigh borhood of any vertex is a strongly regular graph with k' = 2μ'.An incidence system with a set of points P and a set of straight lines ᏸ is called an α partial geometry of order (s, t) (denoted by pG α (s, t)) if each line contains s + 1 points; each point lies on t + 1 lines; any two points lie on at most one line; and, for any antiflag (a, l) ∈ (P, ᏸ), there are precisely α lines that pass through a and intersect l. If α = 1, the geometry is called a generalized quadrangle and is denoted by GQ(s, t). The point graph of a geometry is defined on the set of points P, and two points are adjacent if they lie on a line. The point graph of pG α (s, t) is strongly regular with v = (s + 1) , k = s(t + 1), λ = s -1 + t(α -1), and μ = α(t + 1). A strongly regular graph with such parameters for some positive integers α, s, and t is called a pseudogeometric graph for pG α (s, t).Makhnev has proposed a program for the study of distance regular graphs in which the neighborhoods of vertices are strongly regular graphs with given parame ters. This program has been implemented in the case of strongly regular graphs with the eigenvalue 2 (see [3]).An antipodal distance regular graph Γ of diameter 3 has (see [1]) the intersection array {k, μ(r -1), 1; 1, μ, k} and possesses v = r(k + 1) vertices and the spectrum k b

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Distance-Regular Graphs

Cited by 1,563 publications

References 281 publications

Decentralized Laplacian Eigenvalues Estimation and Collaborative Network Topology Identification

Decentralized Laplacian Eigenvalues Estimation and Collaborative Network Topology Identification

Spherical and Clockwise Spherical Graphs

On distance-regular graphs in which neighborhoods of vertices are strongly regular

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