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Lower and Upper Acyclicities on Unitary Cayley Graphs of Finite Commutative Rings

Authors

Denpong Pongpipat, Nuttawoot Nupo
Type: Article
Publication Date: 2025-05-02
Citations: 0
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6059

Abstract

A unitary Cayley graph $\Gamma_n$ of a finite cyclic ring $\mathbb{Z}_n$ is a graph with vertex set $\mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $\mathbb{Z}_n$ or equivalently, $\gcd(x-y,n)=1$. A nonempty subset $A$ of $\mathbb{Z}_n$ is called an acyclic set of $\Gamma_n$ if a subgraph of $\Gamma_n$ induced by $A$ contains no cycles. The maximum cardinality among the acyclic sets of $\Gamma_n$ is called the upper acyclic number of $\Gamma_n$ and is denoted by $\Lambda(\Gamma_n)$. Moreover, the maximum number $k$ of vertices of $\Gamma_n$ in which every subgraph of $\Gamma_n$ induced by $k$ vertices contains no cycles is called the lower acyclic number of $\Gamma_n$ and denoted by $\lambda(\Gamma_n)$. In this paper, we determine the lower and upper acyclic numbers for unitary Cayley graphs of $\mathbb{Z}_{n}$ and their complements.

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  • European Journal of Pure and Applied Mathematics
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The unitary Cayley graph of upper triangular matrix rings

2024-03-02
Waldemar Hołubowski , Sergiy Kozerenko , Bogdana Oliynyk +1 more
The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an … The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph $C_{T_n (\mathbb{F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{F})$ over a finite field $\mathbb{F}$. We show that $C_{T_n (\mathbb{F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{\frac{kn(n-1)}{2}}$ and $|\mathbb{F}|=p^k$. In particular, if $|\mathbb{F}|=2$, then the graph $C_{T_n (\mathbb{F})}$ has $2^{n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{m,m}$, where $m=2^{\frac{n(n-1)}{2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{T_n (\mathbb{F})}$.
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ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS

2019-09-01
Meysam Rezagholibeigi , A. R. Naghipour
Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ … Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $R$ is a left Artinian ring. We also determine when the graph $Gamma(R)$ is a cycle graph. We prove that if $Gamma(R)congGamma(M_{n}(F))$ then $Rcong M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. We show that if $R$ is a finite commutative semisimple ring and $S$ is a commutative ring such that $Gamma(R)congGamma(S)$, then $Rcong S$. Finally, we obtain the spectrum of $Gamma(R)$, where $R$ is a finite commutative ring.

A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING

2016-07-31
A. R. Naghipour , Meysam Rezagholibeigi
Let R be a finite commutative ring with nonzero identity. We define <TEX>${\Gamma}(R)$</TEX> to be the graph with vertex set R in which two distinct vertices x and y are … Let R be a finite commutative ring with nonzero identity. We define <TEX>${\Gamma}(R)$</TEX> to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of <TEX>${\Gamma}(R)$</TEX> are obtained and the vertex connectivity and the edge connectivity of <TEX>${\Gamma}(R)$</TEX> are given. Finally, by a constructive way, we determine when the graph <TEX>${\Gamma}(R)$</TEX> is Hamiltonian. As a consequence, we show that <TEX>${\Gamma}(R)$</TEX> has a perfect matching if and only if <TEX>${\mid}R{\mid}$</TEX> is an even number.
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On the Unitary Cayley Graph of a Ring

2012-04-16
Dariush Kiani , Mohsen Molla Haji Aghaei
Let $R$ be a ring with identity. The unitary Cayley graph of a ring $R$, denoted by $G_{R}$, is the graph, whose vertex set is $R$, and in which $\{x,y\}$ … Let $R$ be a ring with identity. The unitary Cayley graph of a ring $R$, denoted by $G_{R}$, is the graph, whose vertex set is $R$, and in which $\{x,y\}$ is an edge if and only if $x-y$ is a unit of $R$. In this paper we find chromatic, clique and independence number of $G_{R}$, where $R$ is a finite ring. Also, we prove that if $G_{R} \simeq G_{S}$, then $G_{R/J_{R}} \simeq G_{S/J_{S}}$, where $\rm J_{R}$ and $\rm J_{S}$ are Jacobson radicals of $R$ and $S$, respectively. Moreover, we prove if $G_{R} \simeq G_{M_{n}(F)}$ then $R\simeq M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. Finally, let $R$ and $S$ be finite commutative rings, we show that if $G_{R} \simeq G_{S}$, then $\rm R/ {J}_{R}\simeq S/J_{S}$.
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GENERALIZED CAYLEY GRAPH OF UPPER TRIANGULAR MATRIX RINGS

2016-07-31
Mojgan Afkhami , Seyed Hosein Hashemifar , Kazem Khashyarmanesh
Let R be a commutative ring with the non-zero identity and n be a natural number. <TEX>${\Gamma}^n_R$</TEX> is a simple graph with <TEX>$R^n{\setminus}\{0\}$</TEX> as the vertex set and two distinct … Let R be a commutative ring with the non-zero identity and n be a natural number. <TEX>${\Gamma}^n_R$</TEX> is a simple graph with <TEX>$R^n{\setminus}\{0\}$</TEX> as the vertex set and two distinct vertices X and Y in <TEX>$R^n$</TEX> are adjacent if and only if there exists an <TEX>$n{\times}n$</TEX> lower triangular matrix A over R whose entries on the main diagonal are non-zero such that <TEX>$AX^t=Y^t$</TEX> or <TEX>$AY^t=X^t$</TEX>, where, for a matrix B, <TEX>$B^t$</TEX> is the matrix transpose of B. <TEX>${\Gamma}^n_R$</TEX> is a generalization of Cayley graph. Let <TEX>$T_n(R)$</TEX> denote the <TEX>$n{\times}n$</TEX> upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph <TEX>${\Gamma}^n_{T_n(R)}$</TEX>.
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Upper ideal relation graphs associated to rings

2024-03-07
Barkha Baloda , Jitender Kumar
Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all … Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if there exists a non-unit element $z \in R$ such that the ideals $(x)$ and $(y)$ contained in the ideal $(z)$. In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of $\Gamma_U(R)$, we determine all the non-local finite commutative rings $R$ whose upper ideal relation graph has genus at most $2$. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of $\Gamma_U(R)$ is either $1$ or $2$.
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Unitary Cayley graphs whose Roman domination numbers are at most four

2022-01-02
A. Y. M. Chin , Hamid Reza Maimani , M. R. Pournaki +2 more

A complete classification of perfect unitary Cayley graphs

2024-09-03
Ján Mináč , Tung Nguyen , Nguyêñ Duy Tân
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 … 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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On line upper ideal relation graphs of rings

2024-05-29
Mohd Shariq , Praveen Mathil , Jitender Kumar
The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex … The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the principal ideals $(x)$ and $(y)$ are contained in the principal ideal $(z)$ for some non-unit element $z\in R$. This manuscript characterizes all the Artinian rings $R$ such that the graph $\Gamma_{U}(R)$ is a line graph. Moreover, all the Artinian rings $R$ for which $\Gamma_{U}(R)$ is the complement of a line graph have been described.
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Some Properties of a Generalization of the Unitary Cayley Graphs of a Finite Commutative Ring

2015-01-01
Xiong Tengfe
Let Rbe a finite commutative ring with non-zero identity and U(R)be the unit group of R.Suppose that Gis a multiplicative subgroup of U(R),and Sis a non-empty subset of Gsuch that … Let Rbe a finite commutative ring with non-zero identity and U(R)be the unit group of R.Suppose that Gis a multiplicative subgroup of U(R),and Sis a non-empty subset of Gsuch that S-1={s-1|s∈S}S.Then the vertex set of unitary Cayley graph Cay(R,U(R))is R,and two distinct vertices xand yare adjacent if and only if x-y∈U(R).Γ(R,G,S)is a graph whose vertex set is R,and vertices xand yare adjacent if and only if there exists s∈Ssuch that x+sy∈G.Obviously,if G=U(R),thenΓ(R,G,{-1})is the unitary Cayley graph.By the structure of a finite commutative ring and the theory of group and graph,we study some properties of a generalization of the unitary Cayley graphs of a finite commutative ring.We consider the regularity of a generalization of the unitary Cayley graphsΓ(R,G,{s}),and determine the number of common neighbors of two distinct vertices ofΓ(R,U(R),{s}).In addiction,evaluate the edge chromatic number ofΓ(R,U(R),{s}).

Nordhaus-gaddum type inequalities for tree covering numbers on unitary cayley graphs of finite rings

2022-06-01
Denpong Pongpipat , Nuttawoot Nupo
The unitary Cayley graph $Gamma_n$ of a finite ring $mathbb{Z}_n$ is the graph with vertex set $mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ … The unitary Cayley graph $Gamma_n$ of a finite ring $mathbb{Z}_n$ is the graph with vertex set $mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $mathbb{Z}_n$‎. ‎A family $mathcal{F}$ of mutually edge disjoint trees in $Gamma_n$ is called a tree cover of $Gamma_n$ if for each edge $ein E(Gamma_n)$‎, ‎there exists a tree $Tinmathcal{F}$ in which $ein E(T)$‎. ‎The minimum cardinality among tree covers of $Gamma_n$ is called a tree covering number and denoted by $tau(Gamma_n)$‎. ‎In this paper‎, ‎we prove that‎, ‎for a positive integer $ ngeq 3 $‎, ‎the tree covering number of $ Gamma_n $ is $ displaystylefrac{varphi(n)}{2}+1 $ and the tree covering number of $ overline{Gamma}_n $ is at most $ n-p $ where $ p $ is the least prime divisor of $n$‎. ‎Furthermore‎, ‎we introduce the Nordhaus-Gaddum type inequalities for tree covering numbers on unitary Cayley graphs of rings $mathbb{Z}_n$‎.

On The automorphism groups of $us$-Cayley graphs

2019-01-01
S. Morteza Mirafzal
Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ … Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation), whenever for every $0 \neq g\in G$ if there are $s_1,s_2,s_3, s_4 \in \Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have $\{s_1,s_2 \} = \{s_3,s_4 \}$. We say that a Cayley graph $\Gamma=Cay(G;\Omega)$ is a $us$-$Cayley\ graph$, whenever $G$ is an abelian group and the generating subset $\Omega$ has the property \lq\lq{}$us$\rq\rq{}. In this paper, we show that if $\Gamma=Cay(G;\Omega)$ is a $us$-$Cayley\ graph$, then $Aut(\Gamma)=L(G)\rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $\theta$ of the group $G$ such that $\theta(\Omega)=\Omega$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including M\"{o}bius ladders and $k$-ary $n$-cubes.
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A study of unit graphs and unitary cayley graphs associated with rings

2015-02-01
Huadong Su
In this thesis, we study the unit graph G(R) and the unitary Cayley graph Γ(R) of a ring R, and relate them to the structure of the ring R. Chapter … In this thesis, we study the unit graph G(R) and the unitary Cayley graph Γ(R) of a ring R, and relate them to the structure of the ring R. Chapter 1 gives a brief history and background of the study of the unit graphs and unitary Cayley graphs of rings. Moreover, some basic concepts, which are needed in this thesis, in ring theory and graph theory are introduced. Chapter 2 concerns the unit graph G(R) of a ring R. In Section 2.2, we first prove that the girth gr(G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. Then we determine the rings R with R/J(R) semipotent and with gr(G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr(G(R)) = 3 or 4. The girth of the unit graphs of some ring extensions are also investigated. The focus of Section 2.3 is on the diameter of unit graphs of rings. We prove that diam(G(R)) ∈ {1, 2, 3,∞} for a ring R with R/J(R) self-injective and determine those rings R with diam(G(R)) = 1, 2, 3 or ∞, respectively. It is shown that, for each n ≥ 1, there exists a ring R such that n ≤ diam(G(R)) ≤ 2n. The planarity of unit graphs of rings is discussed in Section 2.4. We completely determine the rings whose unit graphs are planar. In the last section of this chapter, we classify all finite commutative rings whose unit graphs have genus 1, 2 and 3, respectively. Chapter 3 is about the unitary Cayley graph Γ(R) of a ring R. In Section 3.2, it is proved that gr(Γ(R)) ∈ {3, 4, 6,∞} for an arbitrary ring R, and that, for each n ≥ 1, there exists a ring R with diam(Γ(R)) = n. Rings R with R/J(R) self-injective are classified according to diameters of their unitary Cayley graphs. In Section 3.3, we completely characterize the rings whose unitary Cayley graphs are planar. In Section 3.4, we prove that, for each g ≥ 1, there are at most finitely many finite commutative rings R with genus γ(Γ(R)) = g. We also determine all finite commutative rings R with γ(Γ(R)) = 1, 2, 3, respectively. Chapter 4 is about the isomorphism problem between G(R) and Γ(R). We prove that for a finite ring R, G(R) ∼= Γ(R) if and only if either char(R/J(R)) = 2 or R/J(R) = Z2 × S for some ring S.

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

2024-03-08
Tung T. Nguyen , Nguyêñ Duy Tân
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 … 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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A class of well-covered and vertex decomposable graphs arising from rings

2020-04-01
Morteza Vafaei , Abolfazl Tehranian , R. Nikandish
Let $ mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ mathbb {Z}_{n} $ is defined as the graph $ G( … Let $ mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ mathbb {Z}_{n} $ is defined as the graph $ G( mathbb {Z}_{n} ) $ with the vertex set $ mathbb {Z}_{n} $ and two distinct vertices $a,b$ are adjacent if and only if  $a-bin Uleft( mathbb {Z}_{n}right)$, where $ Uleft( mathbb {Z}_{n}right) $ is the set of units of $ mathbb {Z}_{n} $. Let $Gamma ( mathbb {Z}_{n} ) $ be the complement of $ G( mathbb {Z}_{n} )  $. In this paper, we determine the independence number of $ Gamma ( mathbb {Z}_{n} ) $. Also it is proved that $ Gamma ( mathbb {Z}_{n} ) $ is well-covered.  Among other things, we provide condition under which $ Gamma ( mathbb {Z}_{n} ) $ is vertex decomposable.

Integral Cayley Graphs over Dihedral Groups

2016-01-01
Lu Lu , Qiongxiang Huang , Xueyi Huang
In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient … In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over $D_n$ in terms of the Boolean algebra of $\langle a\rangle$, from which we find infinite classes of integral Cayley graphs over $D_n$. In particular, we completely determine all integral Cayley graphs over the dihedral group $D_p$ for a prime $p$.

Chromatic and achromatic numbers of unitary addition Cayley graphs

2024-07-14
Keenan Calhoun , Yesim Demiroglu , Vincent Pigno +1 more
Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if … Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for the clique number and chromatic number of such graphs when $R$ is a finite commutative ring. This includes the special case when $R$ is $\mathbb{Z}_n$, the integers modulo $n$, where these parameters had been found under the assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally, we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case that $n$ is the product of two primes. We prove that the achromatic number of $\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$ are distinct odd primes.
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Total graph of the ring ℤ<sub>n</sub> × ℤ<sub>m</sub>

2014-12-09
Alpesh M. Dhorajia
Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T Γ (R), is the (undirected) graph … Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T Γ (R), is the (undirected) graph with vertices set R. For any two distinct elements x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we obtain certain fundamental properties of the total graph of ℤ n × ℤ m , where n and m are positive integers. We determine the clique number and independent number of the total graph T Γ (ℤ n × ℤ m ).

Commutative Rings Whose Cozero-Divisor Graphs are Unicyclic or of Bounded Degree

2013-12-07
Saieed Akbari , S. Khojasteh
Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all … Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2), ℤ2[x, y]/(x, y)2, ℤ4[x]/(2x, x 2). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′(R)) ∈ {3, 4, ∞}.

State transfer of Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings

2025-02-14
Koushik Bhakta , Bikash Bhattacharjya
This paper focuses on periodicity and perfect state transfer of Grover walks on two well-known families of Cayley graphs, namely, the unitary Cayley graphs and the quadratic unitary Cayley graphs. … This paper focuses on periodicity and perfect state transfer of Grover walks on two well-known families of Cayley graphs, namely, the unitary Cayley graphs and the quadratic unitary Cayley graphs. Let $R$ be a finite commutative ring. The unitary Cayley graph $G_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ is a unit in $R$. We provide a necessary and sufficient condition for the periodicity of the Cayley graph $G_R$. We also completely determine the rings $R$ for which $G_R$ exhibits perfect state transfer. The quadratic unitary Cayley graph $\mathcal{G}_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ or $v-u$ is a square of some units in $R$. It is well known that any finite commutative ring $R$ can be expressed as $R_1\times\cdots\times R_s$, where each $R_i$ is a local ring with maximal ideal $M_i$ for $i\in\{1,...,s\}$. We characterize periodicity and perfect state transfer on $\mathcal{G}_R$ under the condition that $|R_i|/|M_i|\equiv 1 \pmod 4$ for $i\in\{1,...,s\}$. Also, we characterize periodicity and perfect state transfer on $\mathcal{G}_R$, where $R$ can be expressed as $R_0\times\cdots\times R_s$ such that $|R_0|/|M_0|\equiv3\pmod 4$, and $|R_i|/|M_i|\equiv1\pmod4$ for $i\in\{1,..., s\}$, where $R_i$ is a local ring with maximal ideal $M_i$ for $i\in\{0,...,s\}$.
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