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

10.37236/654 ◽  
2011 ◽  
Vol 18 (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.

scholarly journals Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ying Ying Tan ◽  
Xiaoye Liang ◽  
Jack Koolen
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Survey Paper ◽  
Grassmann Graph ◽  
Distance Regular Graphs ◽  
Classical Parameters

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

Classification of the Known Distance-Regular Graphs

1989 ◽  
pp. 193-213 ◽  
Author(s):  
Andries E. Brouwer ◽  
Arjeh M. Cohen ◽  
Arnold Neumaier
Keyword(s):  
Regular Graphs ◽  
Distance Regular Graphs

On Moore graphs

1973 ◽  
Vol 74 (2) ◽  
pp. 227-236 ◽  
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:

scholarly journals Totally bipartite tridiagonal pairs

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.

The classification of distance-regular graphs of type IIB

COMBINATORICA ◽  
1988 ◽  
Vol 8 (1) ◽  
pp. 125-132 ◽  
Author(s):  
P. Terwilliger
Keyword(s):  
Regular Graphs ◽  
Distance Regular Graphs

scholarly journals A symbolic projection of Langton's Ant

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Anahi Gajardo
Keyword(s):  
Point Of View ◽  
Algebraic Characterization ◽  
Transition Rule ◽  
Regular Graphs ◽  
New Approach ◽  
The Family ◽  
International Audience ◽  
Topological Dynamical Systems

International audience The Langton's ant is studied from the point of view of topological dynamical systems. A new approach which associate a subshift to the system is proposed.The transition rule is generalized to the family of bi-regular graphs $\Gamma(k,d)$ and the dependence of the dynamical system on $k$ and $d$ is analyzed. A classification of the $\Gamma (k,d)$ graphs based on the dynamical properties of the subshift is established. Also a hierarchy is defined on the graphs through the subset relation of the respective subshifts. The analysis are worked out by establishing an algebraic characterization of the forbidden words of the subshift.

scholarly journals Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

2021 ◽  
Author(s):  
Masoumeh Koohestani ◽  
◽  
Nobuaki Obata ◽  
Hajime Tanaka ◽  
◽  
...  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gibbs State ◽  
Scaling Limits ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Quantum Central Limit Theorem ◽  
Weak Limits ◽  
Distance Regular Graphs ◽  
Classical Parameters

We determine the possible scaling limits in the quantum central limit theorem with respect to the Gibbs state, for a growing distance-regular graph that has so-called classical parameters with base unequal to one. We also describe explicitly the corresponding weak limits of the normalized spectral distribution of the adjacency matrix. We demonstrate our results with the known infinite families of distance-regular graphs having classical parameters and with unbounded diameter.

scholarly journals Nonexistence of a Class of Distance-Regular Graphs

10.37236/3356 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Yu-pei Huang ◽  
Yeh-jong Pan ◽  
Chih-wen Weng
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Distance Regular Graphs ◽  
Classical Parameters ◽  
Alpha Beta ◽  
Beta 2

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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