Distance-Regular Graph with c 2 &gt; 1 and a 1 = 0 &lt; a 2

Hiraki, Akira

doi:10.1007/s00373-008-0823-7

Cited by 4 publications

(5 citation statements)

References 14 publications

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Section: Recent Results On Graphs With Classical Parametersmentioning

confidence: 99%

“…Triangle-free distance regular graphs with classical parameters have been studied by Pan, Lu, and Weng [527,528,529] and Hiraki [317]. One of the results is that if Γ has classical parameters and [529,317]. Case (i) with D = 3 is uniquely realized by the Witt graph M 23 [78, §11.4B], whereas Huang, Pan, and Weng [343] ruled out case (i) with D ≥ 4.…”

Section: Recent Results On Graphs With Classical Parametersmentioning

confidence: 99%

See 1 more Smart Citation

Distance-regular graphs

van Dam,

Koolen,

Tanaka

2014

Preprint

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Section: Recent Results On Graphs With Classical Parametersmentioning

confidence: 99%

Section: Recent Results On Graphs With Classical Parametersmentioning

confidence: 99%

Distance-regular graphs

van Dam,

Koolen,

Tanaka

2014

Preprint

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“…Throughout this section, let Γ = (X, R) denote a distance-regular graph with diameter D 3, and intersection numbers a 1 = 0, a 2 = 0. Such graphs are also studied in [4,5,6,7,8].…”

Section: The Shape Of Pentagonsmentioning

confidence: 99%

“…The following three lemmas were formulated by A. Hiraki in [4] under an additional assumption c 2 > 1, but this assumption is essentially not used in his proofs. For the sake of completeness, we still provide the proofs.…”

Section: The Shape Of Pentagonsmentioning

confidence: 99%

Nonexistence of a Class of Distance-Regular Graphs

Huang¹,

Pan²,

Weng³

2015

Electron. J. Combin.

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Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show a connection between the $d$-bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng (2009) showed that $(b, \alpha, \beta)= (-2, -2, ((-2)^{D+1}-1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.

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“…There are many results on distance-regular graphs with the property that any pair of vertices x, y is contained in a strongly closed subset with width ∂(x, y); see e.g., [40,18]. In view of the results of §6, in this section we assume Γ has classical parameters and look at the implications of similar existence conditions on descendents.…”

Section: Quantum Matroids and Descendentsmentioning

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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Distance-Regular Graph with c 2 > 1 and a 1 = 0 < a 2

Cited by 4 publications

References 14 publications

Distance-regular graphs

Distance-regular graphs

Nonexistence of a Class of Distance-Regular Graphs

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

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