arXiv subjects

Yusaku Nishimura

Publications and source records attributed to Yusaku Nishimura.

9 recordsLinked to original sources

arXiv2026

Spherical Designs with Infinite Harmonic Strength

In this paper, we study the existence problem for spherical $T$-designs on the $d$-dimensional sphere, where $T$ is an infinite subset of $\mathbb N$. We show that, if $d\ge 2$, then a finite subset of $S^d$ has infinite harmonic strength if and only if it is antipodal. For $d=1$, we show that infinite strength spherical designs are exactly cyclotomic designs, and we characterize their existence in terms of certain $0$-$1$ polynomials. We also prove that the harmonic strength of every infinite strength spherical design has the weak GCD property. Finally, for a given infinite subset $T\subset \mathbb N$ with the weak GCD property, we give a finite procedure to decide whether there exists $X\subset S^1$ such that $\operatorname{Hst}(X)=T$, and apply this criterion to concrete existence and non-existence examples.

math.CO↗

arXiv2026

An algebraic-combinatorial framework for finding the average hitting times in graphs with high regularity

For any given vertices $u$ and $v$ in a graph, the hitting time of a random walk on a finite graph is the number of steps it takes for a random walk to reach vertex $v$ starting at vertex $u$. The expected value of the hitting time is the average hitting time. In this paper, we present an algebraic-combinatorial method for calculating the average hitting time between vertices of finite graphs exhibiting high regularity, along with its applications to multiple graph classes. Our approach exploits a novel connection between maximal-entropy random walks and weight-equitable partitions, providing a unifying framework that strengthens and extends several known results, including Rao's method [Statistics \& Probability Letters, 2013] for computing the hitting time from a vertex to a neighbor under certain symmetries of the starting vertex.

math.CO↗

arXiv2026

A New Approach to Code Smoothing Bounds

Code smoothing is a phenomenon in which an error distribution makes a code statistically close to the uniform distribution over the ambient space. This closeness is measured by the total variation distance. Recently, Debris-Alazard et al.\ introduced a smoothing bound, which is an upper bound on this total variation distance. Although the smoothing bound evaluates how the error distribution smooths a code, this bound applies only to linear codes. In this paper, we generalize this bound to not only linear codes but also specific non-linear codes. While the smoothing bound in previous work was obtained by Fourier analysis over finite abelian groups, we derive this bound using a graph-theoretic approach. To derive the smoothing bound, we consider code smoothing as the mixing of random walks on a specific graph, and use the concept of equitable partitions, which is well-studied in graph theory.

cs.IT↗

arXiv2025

On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings

These days, post-quantum cryptography based on the lattice isomorphism problem has been proposed. Ducas-Gibbons introduced the hull attack, which solves the lattice isomorphism problem for lattices obtained by Construction A from an LCD code over a finite field. Using this attack, they showed that the lattice isomorphism problem for such lattices can be reduced to the lattice isomorphism problem with the trivial lattice $\mathbb{Z}^n$ and the graph isomorphism problem. While the previous work by Ducas-Gibbons only considered lattices constructed by a code over a \textit{finite field}, this paper considers lattices constructed by a code over a \textit{finite ring} $\mathbb{Z}/k\mathbb{Z}$, which is a more general case. In particular, when $k$ is odd, an odd prime power, or not divisible by $4$, we show that the lattice isomorphism problem can be reduced to the lattice isomorphism problem for $\mathbb{Z}^n$ and the graph isomorphism problem.

cs.IT↗

arXiv2025

Spherical Designs on $S^1$ of Finite Harmonic Strength

We study exact harmonic strengths of finite spherical designs on the unit circle. For a nonempty finite set $X\subset S^1$, let $\Hst(X)$ be the set of positive integers $k$ for which the $k$-th complex moment $P_k(X)=\sum_{x\in X}x^k$ vanishes. Equivalently, $X$ is a spherical $T$-design precisely when $T\subset \Hst(X)$. We consider the exact realization problem: given a finite set $T\subset\mathbb N$, determine whether there exists a finite set $X\subset S^1$ such that $\Hst(X)=T$. We prove that every finite $T\subset\mathbb N$ is realizable. More precisely, for each $t\ge 1$ we construct uncountably many five-point sets with $\Hst(X)=\{t\}$, and we prove that no smaller set can have this exact harmonic strength. A product construction then gives, for every finite $T\subset\mathbb N$, a realization with $|X|=5^{|T|}$. We also initiate the associated minimum-size problem $N(T,2)$. We prove $N(\{t\},2)=5$ for all $t\ge1$, determine $N(\{2,3\},2)=5$, and show that the optimal $\{2,3\}$-example is unique up to rotation. Finally, we discuss a rigid seven-point example related to $T=\{2,3,4,10\}$.

math.CO↗

arXiv2025

Universal graph series, chromatic functions, and their index theory

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.

math.CO↗

arXiv2024

The Kneser chromatic function distinguishes trees

R.P. Stanley defined a invariant for graphs called the chromatic symmetric function and conjectured it is complete invariant for trees. Miezaki et al. generalised the chromatic symmetric function and defined the Kneser chromatic functions denoted by $\{X_{K_{\mathbb{N},k}}\}_{k\in\mathbb{N}}$, and rephrase Stanley's conjecture that $X_{K_{\mathbb{N},1}}$ is a complete invariant for trees. This paper shows $X_{K_{\mathbb{N},2}}$ is a complete invariant for trees.

math.CO↗

arXiv2024

Average hitting times in some $f$-equitable graphs

It is known that the average hitting times of simple random walks from any vertex to any other vertex in distance-regular graphs are determined by their intersection array. In this paper, we introduce a new graph classification called $f$-equitable, utilizing both the equitable partition and the function $f$, which represents a generalization of distance-regular graphs. We determine the average hitting times from any vertex to any other vertex in $f$-equitable graphs by using their parameter referred to as the quotient matrix. Furthermore, we prove that there is some function $f$ such that the Cartesian product of two strongly regular graphs is $f$-equitable. We then calculate the quotient matrix for these graphs and determine the average hitting times from any vertex to any other vertex in these graphs. In the same manner, we determine the average hitting times on some generalized Paley graphs.

math.CO↗

arXiv2023

A new approach to pancyclicity of Paley graphs I

Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. We show that the pancyclicity of specific Cayley graphs and the Cartesian product of specific two graphs. As a corollary of these two theorems, we provide a new proof of the pancyclicity of the Paley graph.

math.CO↗