Some spectral and quasi-spectral characterizations of distance-regular graphs

Abiad, Aida; Dam, E.R. van; Fiol, M. A.

doi:10.1016/j.jcta.2016.04.004

Cited by 13 publications

(20 citation statements)

References 35 publications

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“…. , d. In fact, in our case it can be proved that the betas and gammas are negative, in a similar way as in [1,Lemma 2.3]. Also, similar as in the case of the adjacency predistance polynomials, it can be proved…”

Section: Laplacian Predistance and Hoffman Polynomialssupporting

confidence: 81%

The Laplacian spectral excess theorem for distance-regular graphs

Dam

Fiol

2014

Linear Algebra and its Applications

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The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.

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Section: Laplacian Predistance and Hoffman Polynomialssupporting

confidence: 81%

The Laplacian spectral excess theorem for distance-regular graphs

Dam

Fiol

2014

Linear Algebra and its Applications

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“…Abiad, Van Dam, and Fiol [1] proved Theorem 6 independently. 22 Higman-Sims [30, 25] P G(2, q): projective plane, GQ(q, q): generalized quadrangle, GH(q, q): generalized hexagon, q: prime power Remark 6.…”

Section: Minimizing the Second-largest Eigenvaluementioning

confidence: 95%

Linear Programming Bounds for Regular Graphs

Nozaki

2015

Graphs and Combinatorics

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Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some "dual" technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected k-regular graph satisfying g > 2d − 1 has the minimum second-largest eigenvalue of all k-regular graphs of the same size, where d is the number of distinct non-trivial eigenvalues, and g is the girth. The known graphs satisfying g > 2d − 1 are Moore graphs, incidence graphs of regular generalized polygons of order (s, s), triangle-free strongly regular graphs, and the odd graph of degree 4.

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“…Since the diameter is at most t − 1, the graph satisfies g ≥ 2d − 2, where g is the girth and d is the diameter. By g ≥ 2d − 2, the graph becomes a distance-regular graph [1,Theorem 4.4], [29], and it must have the quotient matrix B(k, t, c) for 1 ≤ c < k, or B(k, t − 1, 1) for c = k (see Proposition 4.6 below). Conversely the distance-regular graph with the quotient matrix B(k, t, c) clearly attains the bound M(k, t, c).…”

Section: Upper Bound For Bipartite Graphs With Given Second Eigenvaluementioning

confidence: 99%

A spectral version of the Moore problem for bipartite regular graphs

Cioabă¹,

Koolen²,

Nozaki³

2019

Algebraic Combinatorics

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Let b(k, θ) be the maximum order of a connected bipartite k-regular graph whose second largest eigenvalue is at most θ. In this paper, we obtain a general upper bound for b(k, θ) for any 0 ≤ θ < 2 √ k − 1. Our bound gives the exact value of b(k, θ) whenever there exists a bipartite distance-regular graph of degree k, second largest eigenvalue θ, diameter d and girth g such that g ≥ 2d − 2. For certain values of d, there are infinitely many such graphs of various valencies k. However, for d = 11 or d ≥ 15, we prove that there are no bipartite distance-regular graphs with g ≥ 2d − 2.

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Some spectral and quasi-spectral characterizations of distance-regular graphs

Cited by 13 publications

References 35 publications

The Laplacian spectral excess theorem for distance-regular graphs

The Laplacian spectral excess theorem for distance-regular graphs

Linear Programming Bounds for Regular Graphs

A spectral version of the Moore problem for bipartite regular graphs

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