Mitsugu Hirasaka’s research while affiliated with Pusan National University and other places

In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying $H <N_G(H)<N_G(N_G(H))\unlhd G$ and $|N_G(N_G(H))|=p^3$ where p is a prime. By a corollary of our main result we obtain some inequality which corresponds to the fact $|G:N_G(N_G(H))|\leq p+1$.

Let Δ$\varDelta$ be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum directed out-degree z. The largest possible number v of vertices of Δ$\varDelta$ is given by the following generalization of the Moore bound: v≤(z+t)2+z+1,$\begin{aligned} v\le (z+t)^2+z+1, \end{aligned}$and a digraph attaining this bound is called a Moore digraph. Apart from the case t=1t=1, only three Moore digraphs are known, which are also Cayley graphs. Using computer search, Erskine (J Interconnect Netw, 17: 1741010, 2017) ruled out the existence of further examples of Cayley digraphs attaining the Moore bound for all orders up to 485. We use an algebraic approach to this problem, which goes back to an idea of G. Higman from the theory of association schemes, also known as Benson’s Lemma in finite geometry, and show non-existence of Moore Cayley digraphs of certain orders.

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any $\{1,p\}$-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying H < NG(H)< NG(NG(H)) \trianglelefteq G and |NG(NG(H))|=p^3 where p is a prime, By a corollary of our main result we obtain some inequality which corresponds to the fact |G:NG(NG(H))|≤ p+ 1.

In this paper, we are focusing on {1, 3}-schemes. In my work I am going to prove that {1, 3}-schemes whose thin residues are isomorphic to C 3 C 3 and each relation out of thin residue has valency 3, are finite. REFERENCES 1. S. Bandyopadhyay, P.O. Boykin, V. Roychowdhury, F. Vatan, A new proof for the existence of mutually unbiased bases, Quantum computation and quantum cryptography.

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique we prove a new result: given a prime p, any $\{1,p\}$-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

A colored space is the pair (X,r) of a set X and a function r whose domain is $\binom{X}{2}$. Let (X,r) be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that r(U)=r(f(U)) for each $U\in\binom{Y}{2}$. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in $\binom{X}{2}$ and $\binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens when the equality holds.

Let (X,d) be a finite metric space with $|X|=n$. For a positive integer k we define $A_k(X)$ to be the quotient set of all k-subsets of X by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,\ldots,a_{n})$ is called the \textit{isometric} sequence of (X,d). In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) $a_k=1$ for some k with $2\leq k\leq n-2$; (ii) $a_k=2$ for some k with $4\leq k\leq \frac{1+\sqrt{1+4n}}{2}$; (iii) $a_3=2$; (iv) $a_2=a_3=3$. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the d-dimensional Euclidean space with small $a_3$, which are analogue pro

A permutation group is said to be quasiregular if every its transitive constituent is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with homogeneous components have distributive lattices of normal subgroups.