scholarly journals The Terwilliger Algebra of the Incidence Graphs of Johnson Geometry

10.37236/3751 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Qian Kong ◽  
Benjian Lv ◽  
Kaishun Wang
Keyword(s):  
Terwilliger Algebra ◽  
Johnson Graph ◽  
Incidence Graph ◽  
Johnson Graphs

In 2007, Levstein and Maldonado  computed the Terwilliger algebra of the Johnson graph $J(n,m)$ when $3m\leq n$. It is well known that the halved graphs of the incidence graph $J(n,m,m+1)$ of Johnson geometry are Johnson graphs. In this paper, we determine the Terwilliger algebra of $J(n,m,m+1)$ when $3m\leq n$, give two bases of this algebra, and calculate its dimension.

SCALING LIMIT FOR GIBBS STATES OF JOHNSON GRAPHS AND RESULTING MEIXNER CLASSES

2003 ◽  
Vol 06 (01) ◽  
pp. 139-143 ◽  
Author(s):  
AKIHITO HORA
Keyword(s):  
Fock Space ◽  
Gibbs State ◽  
Scaling Limit ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Johnson Graph ◽  
Meixner Polynomials ◽  
Johnson Graphs ◽  
The One ◽  
Interacting Fock Space

Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.

scholarly journals On the Number of Matroids Compared to the Number of Sparse Paving Matroids

10.37236/4899 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Rudi Pendavingh ◽  
Jorn Van der Pol
Keyword(s):  
Stable Set ◽  
Asymptotic Enumeration ◽  
Stable Sets ◽  
Johnson Graph ◽  
Johnson Graphs ◽  
Other Information ◽  
One To One ◽  
Almost All

It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that $\lim_{n\rightarrow\infty} s_n/m_n = 1$, where $m_n$ denotes the number of matroids on $n$ elements, and $s_n$ the number of sparse paving matroids. In this paper, we show that $$\lim_{n\rightarrow \infty}\frac{\log s_n}{\log m_n}=1.$$ We prove this by arguing that each matroid on $n$ elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on $n$ elements.As a consequence of our result, we find that for all $\beta > \displaystyle{\sqrt{\frac{\ln 2}{2}}} = 0.5887\cdots$, asymptotically almost all matroids on $n$ elements have rank in the range $n/2 \pm \beta\sqrt{n}$.

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

Perfect 3-Colorings of the Johnson Graph J(6, 3)

2020 ◽  
Vol 46 (6) ◽  
pp. 1603-1612
Author(s):  
Mehdi Alaeiyan ◽  
Amirabbas Abedi ◽  
Mohammadhadi Alaeiyan
Keyword(s):  
Johnson Graph
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