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 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.

Central Limit Theorems and Asymptotic Spectral Analysis on Large Graphs

1998 ◽  
Vol 01 (02) ◽  
pp. 221-246 ◽  
Author(s):  
Akihito Hora
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variable ◽  
Spectral Structure ◽  
Regular Graphs ◽  
Large Graphs ◽  
The Central Limit Theorem ◽  
Distance Regular Graphs

Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.

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.

A q‐analog of the quantum central limit theorem for SUq(2)

1992 ◽  
Vol 33 (8) ◽  
pp. 2768-2778 ◽  
Author(s):  
Romuald Lenczewski ◽  
Krzysztof Podgórski
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Quantum Central Limit Theorem

Merging the first and third classes in bipartite distance-regular graphs

2019 ◽  
Vol 12 (07) ◽  
pp. 2050009
Author(s):  
Siwaporn Mamart ◽  
Chalermpong Worawannotai
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Original Graph ◽  
Distance Regular Graphs

Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. We determine when merging the first and third classes in a bipartite distance-regular graph produces a distance-regular graph.

scholarly journals The Random Normal Matrix Model: Insertion of a Point Charge

2021 ◽  
Author(s):  
Yacin Ameur ◽  
Nam-Gyu Kang ◽  
Seong-Mi Seo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Polynomial ◽  
Central Limit ◽  
Point Charge ◽  
Coulomb Gas ◽  
Normal Matrix ◽  
Scaling Limits ◽  
Two Dimensional ◽  
Rotationally Symmetric

AbstractIn this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for $\log |p_{n}(\zeta )|$ log | p n ( ζ ) | where pn is the characteristic polynomial of an n:th order random normal matrix.

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.

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