arXiv subjects

Tung T. Nguyen

Publications and source records attributed to Tung T. Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

Isomorphic gcd-graphs over polynomial rings

Gcd-graphs over the ring of integers modulo $n$ are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We discover that, in both cases, gcd-graphs share many similar and analogous properties. In this article, we extend this line of research further. Among other topics, we explore an analog of a conjecture of So and a weaker version of Sander-Sander, concerning the conditions under which two gcd-graphs are isomorphic or isospectral. We also provide several constructions showing that, unlike the case over $\mathbb{Z}$, it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.

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arXiv2026

Sums of units in finite rings and applications to Cayley graphs

The question of whether a ring is additively generated by its units has been studied from several perspectives in ring theory and algebraic graph theory. In this paper, we investigate this problem for finite rings, not necessarily commutative, and relate it to the connectedness of gcd-graphs, the existence of perfect state transfer, and the solvability of certain equations over finite fields. Additionally, we discuss a generalization of this question in which only certain normalized units are allowed in the generating set. Our work intersects algebra, number theory, and graph theory, and may be of interest to a broad audience.

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arXiv2026

Perfect state transfer on gcd-graphs over a finite Frobenius ring

The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over $\mathbb{Z}$ and cubelike graphs. In this article, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.

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arXiv2026

Cyclotomic factors of rational necklace functions

Necklace polynomials arise in various fields of mathematics, including combinatorics, Lie theory, and Galois theory over finite fields. Their arithmetic properties have been extensively studied in recent years. In this article, we introduce a new class of rational necklace functions that unifies two well-studied families of polynomials: necklace polynomials and Fekete polynomials. We describe several ways in which cyclotomic polynomials appear as factors of these rational necklace functions. Our results unify and generalize various earlier work on necklace polynomials and on Fekete polynomials. In particular, we describe a surprising phenomenon in which certain Galois groups play a hidden role in the appearance of new cyclotomic factors that are not covered by these previous works.

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arXiv2026

On gcd-graphs over matrix rings

Graphs defined over finite rings are well studied in the literature. The study of these graphs benefits from rich connections between several areas of mathematics, including number theory, algebra, combinatorics, and graph theory, and these connections often lead to interesting interactions between algebraic and combinatorial structures. In this article, we investigate gcd-graphs defined over matrix rings with coefficients in finite fields. We show that these graphs exhibit several interesting graph-theoretic properties. Along the way, we also prove some results on the structure of matrix rings, which may be of independent interest.

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arXiv2026

Analytic Properties of Necklace Polynomials

The necklace polynomials \[ M_n(x)=\frac1n\sum_{d\mid n}μ(d)x^{n/d} \] play a central role in discrete mathematics: they count aperiodic necklaces, enumerate monic irreducible polynomials over finite fields, and give the dimensions of homogeneous components of free Lie algebras. Despite their inherently discrete origins, we show that treating $M_n(x)$ as a function of a real variable $x$ unlocks surprising structural properties that answer natural enumerative questions. In this paper, we study $M_n(x)$ as a real-variable function and establish several new analytical and monotonicity properties. We prove that the normalized functions $M_n(x)/x^n$ and their higher normalized derivatives are strictly increasing on $[1,\infty)$. As a consequence, we show that the proportion of irreducible polynomials of fixed degree over $\mathbf F_q$ increases with $q$. We also establish strict growth with respect to the degree $n$ for $x\ge2$. In addition, we determine a sharp threshold for log-convexity: the sequence $\{M_n(x)\}_{n\ge2}$ is uniformly log-convex if and only if $x>8$. These results reveal unexpected analytic structure underlying necklace polynomials and show how real-variable methods can yield new information about discrete enumeration problems. For instance, it is shown that adding one more bead to a sufficiently long necklace will approximately increase the total number of primitive, rotationally distinct configurations by a factor of the number of available colors.

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arXiv2026

On the arithmetic of the join rings over finite fields

Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we have previously introduced and studied certain foundational properties of the join ring $\mathcal{J}_{G_1, G_2, \ldots, G_d}(R)$. This ring bridges two extreme worlds: matrix rings $M_n(R)$ on one end, and group rings $R[G]$ on the other. The construction of this ring was motivated by various problems in graph theory, network theory, nonlinear dynamics, and neuroscience. In this paper, we continue our investigations of this ring, focusing more on its arithmetic properties. We begin by constructing a generalized augmentation map that gives a structural decomposition of this ring. This decomposition allows us to compute the zeta function of the join of group rings. We show that the join of group rings is a natural home for studying the concept of simultaneous primitive roots for a given set of primes. This concept is related to the order of the unit group of the join of group rings. Finally, we characterize the join of group rings over finite fields with the property that the order of every unit divides a fixed number. Remarkably, Mersenne and Fermat primes unexpectedly emerge within the context of this exploration.

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arXiv2026

On $U$-unitary Cayley graphs over finite rings

Graphs defined over a finite ring are well-studied in the literature. Due to their nature, these types of graphs connect several branches of mathematics, including algebra, number theory, matrix theory, and representation theory. In recent work, we studied $U$-unitary Cayley graphs over a finite commutative ring, which unifies several constructions of graphs with arithmetic origins. Among various structural graph-theoretic results on these graphs--such as their connectedness, primeness, and bipartiteness--we show that their spectra can be described via a certain supercharacter theory. Utilizing this spectral description, we are able to find some classes of gcd-graphs that possess perfect state transfer. In this article, we generalize this study to finite non-commutative rings, with a special focus on the case of the matrix rings with coefficients in a finite field. We show, in particular, that gcd-graphs over these matrix rings have no perfect state transfer.

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arXiv2025

On the gcd graphs over polynomial rings

Gcd-graphs over the ring of integers modulo $n$ are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over $\mathbb{Z}.$

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arXiv2025

Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs

We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group $G$ is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where $G$ is a cyclic group. We then use our theory to define and study the concept of a $U$-unitary Cayley graph over a finite commutative ring $R$, where $U$ is a subgroup of the unit group of $R$. Furthermore, when the underlying ring is a Frobenius ring, we show that there is a natural supercharacter theory associated with $U$. By applying the general theory of super-Cayley graphs developed in the first part, we explore various spectral properties of these $U$-unitary Cayley graphs, including their rationality and connections to various arithmetical sums.

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arXiv2025

On divisibility relation graphs

For each positive integer $n$, we define the divisibility relation graph $D_n$ whose vertex set is the set of divisors of $n$, and in which two vertices are adjacent if one is a divisor of the other. This type of graph is a special case of graphs associated with a partial order, which have been widely studied in the literature. In this work, we determine various graph-theoretic invariants of divisibility relation graphs, such as their clique and independence numbers, and their planarity. We also discuss various spectral properties that are discovered by our numerical experiments.

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arXiv2025

On gcd-graphs over finite rings

Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define and establish some foundational properties of gcd-graphs defined over a general finite commutative ring. In particular, we investigate the connectivity and diameter of these graphs. Additionally, when the ring is a finite symmetric $\mathbb{Z}/n$-algebra, we give an explicit description of their spectrum using the theory of Ramanujan sums that gives a unified treatment of various results in the literature.

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arXiv2025

Integral Cayley graphs over a finite symmetric algebra

A graph is called integral if its eigenvalues are integers. In this article, we provide the necessary and sufficient conditions for a Cayley graph over a finite symmetric algebra $R$ to be integral. This generalizes the work of So who studies the case where $R$ is the ring of integers modulo $n.$ We also explain some number-theoretic constructions of finite symmetric algebras arising from global fields, which we hope could pave the way for future studies on Paley graphs associated with a finite Hecke character.

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arXiv2024

Joins of normal matrices, their spectrum, and applications

Motivated by studies of oscillator networks, we study the spectrum of the join of several normal matrices with constant row sums. We apply our results to compute the characteristic polynomial of the join of several regular graphs. We then use this theorem to study several problems in spectral graph theory. In particular, we provide some simple constructions of Ramanujan graphs and give new proofs for some theorems in the classical book of Cvetković, Rowlinson, and Slobodan.

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arXiv2024

A complete classification of perfect unitary Cayley graphs

Due to their elegant and simple nature, unitary Cayley graphs have been an active research topic in the literature. These graphs are naturally connected to several branches of mathematics, including number theory, finite algebra, representation theory, and graph theory. In this article, we study the perfectness property of these graphs. More precisely, we provide a complete classification of perfect unitary Cayley graphs associated with finite rings.

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arXiv2024

On certain properties of the $p$-unitary Cayley graph over a finite ring

In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to be prime. In this paper, we continue this line of research. Specifically, we investigate some basic properties of certain $p$-unitary Cayeley graphs associated with a finite commutative ring. In particular, under some mild conditions, we provide the necessary and sufficient conditions for this graph to be prime.

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arXiv2024

On prime Cayley graphs

The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime?

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arXiv2023

Fekete polynomials of principal Dirichlet characters

Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly simpler yet rich variant: the Fekete polynomial $F_n(x) = \sum_{a=1}^n χ_n(a) x^a$ associated to a principal Dirichlet character $χ_n$ of modulus $n$. We investigate the cyclotomic factors of $F_n$ and conjecturally describe all of them. One interesting observation from our computations is that the non-cyclotomic part $f_n$ of $F_n(x)/x$ seems to be always irreducible. We study this factor closely in the special case that $n$ is a product of two odd primes, proving separability in specific cases, and studying its coefficients and special values. Combining these theoretical results with computational evidence lets us identify the Galois group of $f_n$ for small $n$, and raises precise questions in general.

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