The spectra of the local graphs of the twisted Grassmann graphs

Bang, Sejeong; Fujisaki, Tatsuya; Koolen, Jack H.

doi:10.1016/j.ejc.2008.07.016

Cited by 25 publications

(30 citation statements)

References 5 publications

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“…Let T = T (x) denote the subalgebra of Mat V (C) generated by M and M * . We call T the Note that T may depend on the choice of the base vertex (see [8]).…”

Section: Refining K 2tmentioning

confidence: 99%

Problems on Graphs with Fixed Smallest Eigenvalue

Koolen

Yang

2020

Algebra Colloq.

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For strongly regular graphs with parameters (v, k, a, c) and fixed smallest eigenvalue −λ, Neumaier [5] gave two bounds of c by using algebraic property of strongly regular graphs. We define a new class of regular graphs called sesqui-regular graphs, which are amply regular graphs without parameter a. We prove that if a sesqui-regular graph * J.H. with parameters (v, k, c) has very large k and fixed smallest eigenvalue −λ, then c ≤ λ 2 (λ − 1) or v − k − 1 ≤ (λ−1) 2 4 + 1 holds.

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“…Let T = T (x) denote the subalgebra of Mat V (C) generated by M and M * . We call T the Note that T may depend on the choice of the base vertex (see [8]).…”

Section: Refining K 2tmentioning

confidence: 99%

Problems on Graphs with Fixed Smallest Eigenvalue

Koolen

Yang

2020

Algebra Colloq.

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“…The Terwilliger (or subconstituent) algebra T = T (x) with respect to x is the matrix algebra generated by M and M * (x), see [15]. (Note that T (x) may depend on the base vertex x, as for example it is the case for the twisted Grassmann graphs, see [2].) That this algebra is semi-simple is a standard observation.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs

Gavrilyuk

Koolen

2015

Linear Algebra and its Applications

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Let Γ be a Q-polynomial distance-regular graph with diameter at least 3. Terwilliger (1993) implicitly showed that there exists a polynomial, say T (λ) ∈ C[λ], of degree 4 depending only on the intersection numbers of Γ and such that T (η) ≥ 0 holds for any non-principal eigenvalue η of the local graph Γ(x) for any vertex x ∈ V (Γ).We call T (λ) the Terwilliger polynomial of Γ. In this paper, we give an explicit formula for T (λ) in terms of the intersection numbers of Γ and its dual eigenvalues. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least 3 are known.

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“…Then |Y | = m + 1. On the other hand, 1 2 H(n, 2) has maximal clique sizes 4 and n [15,Theorem 14], from which it follows that n = 2d.…”

Section: Halved Cubesmentioning

confidence: 99%

Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs

Tanaka¹

2011

Electron. J. Combin.

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We study Q-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width w and dual width w * satisfy w + w * = d, where d is the diameter of the graph. We show among other results that a nontrivial descendent with w 2 is convex precisely when the graph has classical parameters. The classification of descendents has been done for the 5 classical families of graphs associated with short regular semilattices. We revisit and characterize these families in terms of posets consisting of descendents, and extend the classification to all of the 15 known infinite families with classical parameters and with unbounded diameter.

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The spectra of the local graphs of the twisted Grassmann graphs

Cited by 25 publications

References 5 publications

Problems on Graphs with Fixed Smallest Eigenvalue

Problems on Graphs with Fixed Smallest Eigenvalue

The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs

Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs

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