Classification of the Known Distance-Regular Graphs

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

The Electronic Journal of Combinatorics ◽

10.37236/654 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Hajime Tanaka

Keyword(s):

Point Of View ◽

Regular Graphs ◽

Minimal Width ◽

Distance Regular Graphs ◽

Classical Parameters

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 $w\geq 2$ 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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On Moore graphs

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100048015 ◽

1973 ◽

Vol 74 (2) ◽

pp. 227-236 ◽

Cited By ~ 92

Author(s):

R. M. Damerell

Keyword(s):

Regular Graphs ◽

Distance Regular Graphs ◽

Image Position

In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove that there are no Moore graphs (other than polygons) of diameter ≥ 3. An independent proof of this result has been given by Barmai and Ito(1). Taken with the result of (4), this shows that the only possible Moore graphs are the following:

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Totally bipartite tridiagonal pairs

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5029 ◽

2021 ◽

Vol 37 ◽

pp. 434-491

Author(s):

Kazumasa Nomura ◽

Paul Terwilliger

Keyword(s):

First Principles ◽

Modular Group ◽

Regular Graphs ◽

Matrix Representations ◽

Tridiagonal Pairs ◽

Distance Regular Graphs ◽

Tridiagonal Pair ◽

Tridiagonal Systems ◽

Special Case

There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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