Quadratic unitary Cayley graphs of finite commutative rings

The purpose of this paper is to study spectral properties of a family of Cayley graphs on finite commutative rings. Let $R$ be such a ring and $R^\times$ its set of units. Let $Q_R=\{u^2: u\in R^\times\}$ and $T_R=Q_R\cup(-Q_R)$. We define the quadratic unitary Cayley graph of $R$, denoted by $\mathcal{G}_R$, to be the Cayley graph on the additive group of $R$ with respect to $T_R$; that is, $\mathcal{G}_R$ has vertex set $R$ such that $x, y \in R$ are adjacent if and only if $x-y\in T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1\times R_2\times\cdots\times R_s$, where each $R_i$ is a local ring with maximal ideal $M_i$. Let $R_0$ be a local ring with maximal ideal $M_0$ such that $|R_0|/|M_0| \equiv 3\,(\mod\,4)$. We determine the spectra of $\mathcal{G}_R$ and $\mathcal{G}_{R_0\times R}$ under the condition that $|R_i|/|M_i|\equiv 1\,(\mod\,4)$ for $1 \le i \le s$. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan

Spectral properties of unitary Cayley graphs of finite commutative rings

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set ${{a,b}:a,b\in R, a-b\in R^\times}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph

Enumerating Cliques in Direct Product Graphs

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erd\H{o}s and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs.Comment: 5 pages, 1 figur

Eigenvalues of Cayley graphs

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs

Longest Induced Cycles on Cayley Graphs

In this paper we study the length of the longest induced cycle in the unitary Cayley graph $X_n = Cay(\mathbb Z_n; U_n)$, where $U_n$ is the group of units in $\mathbb Z_n$. Using residues modulo the primes dividing $n$, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing $n$, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in $X_n$. We also see that if $n$ has $r$ distinct prime divisors, $X_n$ always contains an induced cycle of length $2^r+2$, improving the $r \ln r$ bound of Berrezbeitia and Giudici. Moreover, we extend our results for $X_n$ to conjunctions of complete $k_i$-partite graphs, where $k_i$ need not be finite, and also to unitary Cayley graphs on any quotient of a Dedekind domain.Comment: 16 page

On the spectrum of unitary finite-Euclidean graphs

We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even

Walks on Unitary Cayley Graphs and Applications

In this paper, we determine an explicit formula for the number of walks in $X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, the unitary Cayley Graphs of order $n$, between any pair of its vertices. With this result, we give the number of representations of a fixed residue class $\bmod{}n$ as the sum of $k$ units of $\mathbb{Z}_n$

Domination Parameters of the Unitary Cayley Graph of Z/nZ\mathbb{Z}/n\mathbb{Z}

The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a family of infinitely many integers $n$ such that $\gamma_t(X_n) \leq g(n) - 2$, where $\gamma_t$ denotes the total domination number and $g$ denotes the Jacobsthal function. We determine the irredundance number, domination number, and lower independence number of certain direct products of complete graphs and give bounds for these parameters for any direct product of complete graphs. We provide upper bounds on the size of irredundant sets in direct products of balanced, complete multipartite graphs which are asymptotically correct for the unitary Cayley graphs of integers with a bounded smallest prime factor.Comment: 18 pages, 1 figur

Generalized Linear Cellular Automata in Groups and Difference Galois Theory

Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group. We study those systems from the stand point of the Galois theory of difference equations and discrete Fourier transform.Comment: 26 pages, 4 figure

Domination and Upper Domination of Direct Product Graphs

The unitary Cayley graph of $\mathbb{Z} /n \mathbb{Z}$, denoted $X_{\mathbb{Z} / n \mathbb{Z}}$, has vertices $0,1, \dots, n-1$ with $x$ adjacent to $y$ if $x-y$ is relatively prime to $n$. We present results on the tightness of the known inequality $\gamma(X_{\mathbb{Z} / n \mathbb{Z}})\leq \gamma_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)$, where $\gamma$ and $\gamma_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal's function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $\gamma(X_{\mathbb{Z} / n \mathbb{Z}})\leq\gamma_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)-1$. Extending work of Meki\v{s}, we give lower bounds for the domination numbers of direct products of complete graphs. We also present a simple conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs and prove the conjecture in certain cases. We end with some open problems.Comment: 16 pages, 1 figur