186 citations · 467 across the 31 of their papers we have counts for
24 papers · 1 filter
On the spherical design properties of a - and -polynomial association scheme
Jesse Lansdown, William J. Martin, Akihiro Munemasa +2
We show that the strength as a spherical design of the spherical embedding of a - and -polynomial association scheme with at least three classes with respect to a -polynom…
Characterizing graphs with fully positive semidefinite -matrices
Hajime Tanaka
For , the -matrix of a connected simple graph is , where denotes the path-length distance. Descr…
The real equiangular tight frames obtained from rank graphs
Eiichi Bannai, Etsuko Bannai, Chin-Yen Lee +2
We present all nontrivial real equiangular tight frames in obtained as spherical embeddings of primitive rank graphs on vertices, and those…
A duality of scaffolds for translation association schemes
Xiaoye Liang, Ying-Ying Tan, Hajime Tanaka +1
Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and wi…
Spatial Search on Johnson Graphs by Continuous-Time Quantum Walk
Hajime Tanaka, Mohamed Sabri, Renato Portugal
Spatial search on graphs is one of the most important algorithmic applications of quantum walks. To show that a quantum-walk-based search is more efficient than a random-walk-based…
Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters
Masoumeh Koohestani, Nobuaki Obata, Hajime Tanaka
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 par…
The Terwilliger algebra of the twisted Grassmann graph: the thin case
Hajime Tanaka, Tao Wang
The Terwilliger algebra of a finite connected simple graph with respect to a vertex is the complex semisimple matrix algebra generated by the adjacency matrix of…
Tight relative -designs on two shells in hypercubes, and Hahn and Hermite polynomials
Eiichi Bannai, Etsuko Bannai, Hajime Tanaka +1
Relative -designs in the -dimensional hypercube are equivalent to weighted regular -wise balanced designs, which generalize combinatorial - des…
The Independence Number of the Orthogonality Graph in Dimension
Ferdinand Ihringer, Hajime Tanaka
We determine the independence number of the orthogonality graph on -dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in qu…
Asymptotic joint spectra of Cartesian powers of strongly regular graphs and bivariate Charlier-Hermite polynomials
John Vincent S. Morales, Nobuaki Obata, Hajime Tanaka
Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph which is the Cartesian power of the…
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
Jae-Ho Lee, Hajime Tanaka
Let be a dual polar graph with diameter , having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{…
An Assmus-Mattson theorem for codes over commutative association schemes
John Vincent S. Morales, Hajime Tanaka
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with classes). This in particular general…
A semidefinite programming approach to a cross-intersection problem with measures
Sho Suda, Hajime Tanaka, Norihide Tokushige
We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ra…
Distance-regular graphs
Edwin R. van Dam, Jack H. Koolen, Hajime Tanaka
This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of s…
A cross-intersection theorem for vector spaces based on semidefinite programming
Sho Suda, Hajime Tanaka
Let and be families of - and -dimensional subspaces, respectively, of a given -dimensional vector space over a finite field .…
On relative -designs in polynomial association schemes
Eiichi Bannai, Etsuko Bannai, Sho Suda +1
Motivated by the similarities between the theory of spherical -designs and that of -designs in -polynomial association schemes, we study two versions of relative -desig…
A note on super Catalan numbers
Evangelos Georgiadis, Akihiro Munemasa, Hajime Tanaka
We show that the super Catalan numbers are special values of the Krawtchouk polynomials by deriving an expression for the super Catalan numbers in terms of a signed set.
The Erdős-Ko-Rado theorem for twisted Grassmann graphs
Hajime Tanaka
We present a "modern" approach to the Erdős-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by van Dam and…
Vertex subsets with minimal width and dual width in -polynomial distance-regular graphs
Hajime Tanaka
We study -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 and du…
Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes
Hajime Tanaka, Rie Tanaka
Suzuki (1998) showed that an imprimitive Q-polynomial association scheme with first multiplicity at least three is either Q-bipartite, Q-antipodal, or with four or six classes. The…
Arithmetic completely regular codes
J. H. Koolen, W. S. Lee, W. J. Martin +1
In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property tha…
Commutative association schemes
William J. Martin, Hajime Tanaka
Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in…
Group-case commutative association schemes and their character tables
Sung Y. Song, Hajime Tanaka
Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources…
New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
Hajime Tanaka
We use the Terwilliger algebra to provide a new approach to the Assmus-Mattson theorem. This approach also includes another proof of the minimum distance bound shown by Martin as w…