scholarly journals The twisted Grassmann graph is the block graph of a design

2011 ◽  
Vol 12 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Akihiro Munemasa ◽  
Vladimir Tonchev
Keyword(s):  
Block Graph ◽  
Grassmann Graph ◽  
Twisted Grassmann Graph

scholarly journals The Terwilliger Algebra of the Twisted Grassmann Graph: the Thin Case

10.37236/9873 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Hajime Tanaka ◽  
Tao Wang
Keyword(s):  
Automorphism Group ◽  
Adjacency Matrix ◽  
Matrix Algebra ◽  
Simple Graph ◽  
Characteristic Vector ◽  
Terwilliger Algebra ◽  
Grassmann Graph ◽  
Complex Semisimple ◽  
Vertex Set ◽  
Twisted Grassmann Graph

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.

Block-Graph Width

2009 ◽  
pp. 150-157 ◽  
Author(s):  
Maw-Shang Chang ◽  
Ling-Ju Hung ◽  
Ton Kloks ◽  
Sheng-Lung Peng
Keyword(s):  
Block Graph

scholarly journals Perfect 2-Colorings of the Grassmann Graph of Planes

10.37236/8672 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Stefaan De Winter ◽  
Klaus Metsch
Keyword(s):  
Infinite Family ◽  
Grassmann Graph ◽  
The Family

We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG$(n,q)$, $n\ge 5$ odd, and show that the members of the family are the smallest possible examples if $n\ge 9$ or $q\ge 25$.

scholarly journals On the binomial edge ideals of block graphs

2016 ◽  
Vol 24 (2) ◽  
pp. 149-158 ◽  
Author(s):  
Faryal Chaudhry ◽  
Ahmet Dokuyucu ◽  
Rida Irfan
Keyword(s):  
Block Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Initial Ideal ◽  
Block Graphs

Abstract We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.

Inverse of the distance matrix of a bi-block graph

2015 ◽  
Vol 64 (8) ◽  
pp. 1509-1517 ◽  
Author(s):  
Yaoping Hou ◽  
Yajing Sun
Keyword(s):  
Distance Matrix ◽  
Block Graph

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.

scholarly journals On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

10.37236/7689 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Jürgen Herzog ◽  
Giancarlo Rinaldo
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Block Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Block Graphs ◽  
Extremal Betti Number

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

scholarly journals On the metric dimension of Grassmann graphs

2012 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Robert F. Bailey ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
60Th Birthday ◽  
Metric Dimension ◽  
Special Issue ◽  
Resolving Set ◽  
Grassmann Graph ◽  
Algorithms And Complexity ◽  
International Audience

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.

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