arXiv subjects

Kiyoto Yoshino

Publications and source records attributed to Kiyoto Yoshino.

14 recordsLinked to original sources

arXiv2026

Sets of equiangular lines in dimension $18$ constructed from $A_5^3 \oplus A_1^4$

In 2023, Greaves, Syatriadi, and Yatsyna found a set of $57$ equiangular lines in $\mathbb{R}^{18}$, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of $57$ equiangular lines in $\mathbb{R}^{18}$ as affine equiangular sets in an integral overlattice of $A_9^2 \oplus A_1$. In this paper, we construct further sets of $57$ equiangular lines in $\mathbb{R}^{18}$ from Latin squares of order $6$ and Pasch configurations, realized as affine equiangular sets in an integral overlattice of $A_5^3 \oplus A_1^4$. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.

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arXiv2026

Irreducible rootless unimodular lattices generated by vectors of norm $3$

The classical ADE classification implies that $E_8$ is the unique irreducible unimodular Euclidean lattice generated by vectors of norm $2$. In sharp contrast, for every positive integer $m$, we construct an irreducible odd unimodular lattice of rank $24m$ and minimum norm $3$ that is generated by its minimal vectors.

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arXiv2026

The Lemmens-Seidel conjecture for base size $5$

In 2020, Lin and Yu claimed to prove the so-called Lemmens--Seidel conjecture for base size $5$. However, their proof has a gap. In this paper, we prove the conjecture for base size $5$ using the pillar method. We also show that the sets of $57$ equiangular lines with common angle $\arccos(1/5)$ in dimension $18$ found by Greaves et~al. \ in 2021 are indeed counterexamples to one of Lin and Yu's claims.We prove this by answering the question posed by Greaves et~al. \ in 2021 in the negative.They asked whether these sets are contained in the unique set of $276$ equiangular lines with common angle $\arccos(1/5)$ in dimension $23$. Furthermore, we show that these sets are strongly maximal. This gives a negative answer to the question posed by Cao et~al.\ in 2021. They asked whether the unique set of $276$ equiangular lines with common angle $\arccos(1/5)$ in dimension $23$ is the unique strongly maximal set of equiangular lines with common angle $\arccos(1/5)$.

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arXiv2026

Arc search in graphs via Szegedy walks

This paper studies the search for a single arc in a graph using the Szegedy walk. Arc search can be interpreted as finding a quantum particle not only in its position but also with a specific internal state. The quantum walk employed in this study is essentially the model proposed by Segawa and Yoshie for the purpose of edge search. First, we investigate how the symmetry of a graph is reflected in its time evolution matrix, and provide a sufficient condition under which the success probability of the search is independent of the marked arc. In particular, we prove that if a graph is arc-transitive, the success probability is independent of the choice of the marked arc. Next, we analyze path and cycle graphs and show that the quantum search is ineffective for these graphs, whereas it performs well for complete bipartite graphs $K_{n,n}$. These results provide a theoretical foundation for studying arc and edge searches on various graphs, while also suggesting new problems concerning the eigenvalue analysis of edge-signed graphs in spectral graph theory.

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arXiv2026

Strongly regular and strongly walk-regular graphs that admit perfect state transfer

We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}$, and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.

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arXiv2025

Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$

In 2023, Greaves et~al.\ constructed several sets of 57 equiangular lines in dimension 18. Using the concept of switching root introduced by Cao et~al.\ in 2021, these sets of equiangular lines are embedded in a lattice of rank 19 spanned by norm 3 vectors together with a switching root. We characterize this lattice as an overlattice of the root lattice $A_9\oplus A_9\oplus A_1$, and show that there are at least $246896$ sets of 57 equiangular lines in dimension $18$ arising in this way, up to isometry. Additionally, we prove that all of these sets of equiangular lines are strongly maximal. Here, a set of equiangular lines is said to be strongly maximal if there is no set of equiangular lines properly containing it even if the dimension of the underlying space is increased. Among these sets, there are ones with only six distinct Seidel eigenvalues.

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arXiv2025

Enumeration of sets of equiangular lines with common angle $\arccos(1/3)$

In 2018, Szöllősi and Östergård used a computer to enumerate sets of equiangular lines with common angle $\arccos(1/3)$ in dimension $7$. They observed that the numbers $ω(n)$ of sets of $n$ equiangular lines with common angle $\arccos(1/3)$ in dimension $7$ are almost symmetric around $n=14$. In this paper, we prove without a computer that the numbers $ω(n)$ are indeed almost symmetric by considering isometries from root lattices of rank at most $8$ to the root lattice $\sE_8$ of rank $8$ and type $E$. Also, they determined the number $s(n)$ of sets of $n$ equiangular lines with common angle $\arccos(1/3)$ for $n \leq 13$. We construct all the sets of equiangular lines with common angle $\arccos(1/3)$ in dimension greater than $7$ from root lattices of type $A$ or $D$ with the aid of switching roots. As an application, we determine the number $s(n)$ for every positive integer $n$.

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arXiv2024

Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks

We completely characterize circulant graphs with valency up to $4$ that admit perfect state transfer. Those of valency $3$ do not admit it. On the other hand, circulant graphs with valency $4$ admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.

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arXiv2023

Regular graphs to induce even periodic Grover walks

The interest of this paper is a characterization of graphs that induce periodic Grover walks with given periods. In previous studies, Yoshie has shown that the only graphs that induce odd periodic Grover walks are cycle graphs. However, this problem is largely unsolved for even periods. In this study, we show that regular graphs that induce $2l$-periodic Grover walks are also cycle graphs in most cases, where $l$ is an odd integer. The proof uses Galois theory.

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arXiv2022

A quantum searching model finding one of the edges of a subgraph in a complete graph

Some of the quantum searching models have been given by perturbed quantum walks. Driving some perturbed quantum walks, we may quickly find one of the targets with high probability. In this paper, we construct a quantum searching model finding one of the edges of a given subgraph in a complete graph. How to construct our model is that we label the arcs by $+1$ or $-1$, and define a perturbed quantum walk by the sign function on the set of arcs. After that, we detect one of the edges labeled $-1$ by the induced sign function as fast as possible. This idea was firstly proposed by Segawa et al. in 2021. They only addressed the case where the subgraph forms a matching, and obtained by a combinatorial argument that the time of finding one of the edges of the subgraph is quadratically faster than a classical searching model. In this paper, we show that the model is valid for any subgraph, i.e., we obtain by spectral analysis a quadratic speed-up for finding one of the edges of the subgraph in a complete graph.

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arXiv2022

Constellations in prime elements of number fields

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.

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arXiv2021

Embedding theory of lattices and its application for $2$-integrable lattices

For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$ and determinant $7$ in 1989. We find two more ones of rank $12$ and determinant $15$. Then we introduce a method of embedding a given lattice into a unimodular lattice, which plays a key role in proving minimality of non $2$-integrable lattices and finding candidates for non $2$-integrable lattices.

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arXiv2021

Maximality of Seidel matrices and switching roots of graphs

In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.

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arXiv2020

The uniqueness of covers for widely generalized line graphs

As a natural generalization of line graphs, Hoffman line graphs were defined by Woo and Neumaier. Especially, Hoffman line graphs are closely related to the smallest eigenvalue of graphs, and the uniqueness of strict covers of a Hoffman line graph plays a key role in such a study. In this paper, we prove a theorem for the uniqueness of strict covers under a condition which can be checked in finite time. Our result gives a generalization and a short proof for the main part of [Ars Math.~Contemp. \textbf{1} (2008) 81--98].

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