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جستجوی مقالات مرتبط با کلیدواژه

Cayley graph

در نشریات گروه ریاضی
تکرار جستجوی کلیدواژه Cayley graph در نشریات گروه علوم پایه
  • Afsaneh Khalilipour, Modjtaba Ghorbani *, Majid Arezoomand

    In this study, we review two significant topics: perfect State Transfer (PST) and Pretty Good State Transfer (PGST). These concepts involve designing interactions within a chain of spins on graph structures of networks, enabling a quantum state initially placed at one end to be perfectly or pretty transferred to the opposite end within a specified timeframe. PST and PGST play crucial roles in applications such as quantum information processing, quantum communication networks, and quantum chemistry

    Keywords: Perfect State Transfer, Pretty Good State Transfer, Cayley Graph
  • Martin Knor, Riste Škrekovski, Tomáš Vetrík *
    For $n \ge 2t+1$ where $t \ge 1$, the circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, v_2, \dots , v_{n-1}$ and the edges $v_i v_{i+1}$, $v_i v_{i+2}, \dots , v_i v_{i + t}$, where $i = 0, 1, 2, \dots , n-1$, and the subscripts are taken modulo $n$. We prove that the metric dimension ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 1$ for $t \ge 5$, where the equality holds if and only if $t = 5$ and $n = 13$. Thus ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 2$ for $t \ge 6$. This bound is sharp for every $t \ge 6$.
    Keywords: Cayley Graph, Distance, Resolving Set
  • Samira Fallahpour *, Mohammadreza Salarian

    It has been conjectured there is a Hamiltonian cycle in every Cayley graph. Interest in this and other closely related questions has grown in the past few years. There have been many papers on the topic, but it is still an open question whether every connected Cayley graph has a Hamiltonian cycle. In this paper, we survey the results, techniques, and open problems in the field.

    Keywords: Hamiltonian Cycle, Cayley Graph
  • E. Vatandoost *
    In this paper the zero forcing number as well as propagation time of $Cay(G,\Omega),$ where $G$ is a finite group and $\Omega \subset G \setminus \lbrace 1 \rbrace$ is an inverse closed generator set of $G$ is studied. In particular, it is shown that the propagation time of $Cay(G,\Omega)$ is at most two for some special generators.
    Keywords: Zero Forcing Number, Propagation Time, Cayley Graph
  • Afsane Khalilipour *, Modjtaba Ghorbani
    Perfect state transfer (\textit{PST}) on graphs due to their significant applications in quantum information processing and quantum computations. In the present work, we establish a characterization of Cayley graphs over $U_{6n}$ group, having \textit{PST}.
    Keywords: Linear Transformation, Perfect State Transfer, Matrix Representation, Cayley Graph
  • Mehdi Alaeiyan *, Masoumeh Akbarizadeh, Zahra Heydari

    Let G be a finite group and S be a subset of G such that 1G ̸∈ S and S −1 = S. The Cayley graph Σ = Cay(G, S) on G with respect to S is the graph with the vertex set G such that, for §, † ∈ G, the pair (§, †) is an arc in Cay(G, S) if and only if †§−1 ∈ S. The graph Σ is said to be arc-transitive if its full automorphism group Aut(Σ) is transitive on its arc set. In this paper we give a classification for arc-transitive Cayley graphs with valency six on finite abelian groups which are non-normal. Moreover, we classify all normal Cayley graphs on non-cyclic abelian groups with valency 6.

    Keywords: Cayley graph, normal Cayley graph, arctransitive graph
  • Sh. Chokani, F. Movahedi *, S. M. Taheri
    ‎Let $\Gamma$ be a finite group and $S$ be a non-empty subset of $\Gamma$‎. ‎A Cayley graph of the group $\Gamma$‎, ‎denoted by $Cay(\Gamma‎, ‎S)$ is defined as a simple graph that its vertices are the elements of $\Gamma$ and two vertices $u$ and $v$ are adjacent if $uv^{-1} \in \Gamma$. ‎The minimum edge dominating energy of Cayley graph $Cay(\Gamma‎, ‎S)$ is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph $Cay(\Gamma‎, ‎S)$‎. ‎In this paper‎, ‎we estimate the minimum edge dominating energy of the Cayley graphs for the finite group $S_n$‎.
    Keywords: Minimum edge dominating energy, eigenvalue, Cayley graph, Finite group
  • Afsaneh Khalilipour, Modjtaba Ghorbani *
    Perfect state transfer (\textit{PST}) on graphs due to their significant applications in quantum information processing and quantum computations. In the present work, we establish a characterization of Cayley graphs over $U_{6n}$ group, having \textit{PST}.
    Keywords: Linear transformation, perfect state transfer, matrix representation, Cayley graph
  • Shahram Mehry *
    Let $\Gamma$ be a $k-$regular graph with the second maximum  eigenvalue $\lambda$. Then  $\Gamma$ is said o be Ramanujan graph if $\lambda\leq 2\sqrt{k-1}.$ Let $G$ be a finite group  and $\Gamma=Cay(G,S)$ be a Cayley graph related to $G$.  The aim of this paper is to investigate the Ramanujan Cayley graphs of sporadic groups.
    Keywords: sporadic group, character table, Cayley graph, eigenvalue
  • Sharife Chokani, Fateme Movahedi *, Seyyed Mostafa Taheri
    The minimum edge dominating energy of a graph $G$ is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of $G$. In this paper, for some finite symmetric groups $\Gamma$ and subset $S$ of $\Gamma$, the minimum edge dominating energy of the Cayley graph of the group $\Gamma$, denoted by $Cay(\Gamma, S)$, is investigated.
    Keywords: Cayley graph, Eigenvalue, Minimum edge dominating energy, Symmetric group
  • Modjtaba Ghorbani, Aziz Seyyedhadi, Farzaneh Nowroozi-Larki

    A graph is called one-regular if its full automorphism group acts regularly on the set of arcs. In this paper, we classify all connected one-regular graphs of valency 4 of order $p^2q^2$, where $p>q$ are prime numbers. We also prove that all such graphs are Cayley graphs.

    Keywords: one-regular graph, symmetric graph, Cayley graph
  • مجید آرزومند*

    در این مقاله، ماتریس فاصله و چند جمله ای مشخصه ی یک گراف کیلی روی گروه متناهی G بر حسب نمایش های تحویل ناپذیر گروه G بیان می شوند. فرمول های دقیقی برای مقادیر ویژه ی ماتریس فاصله ی گراف های کیلی مکعبی روی گروه های آبلی و برخی گراف های شناخته شده ی دیگر ارایه می دهیم. خانواده ی نامتناهی از گراف های کیلی که تمام مقادیر ویژه ی ماتریس فاصله ی آن ها اعداد صحیحی هستند، معرفی می کنیم. ثابت می کنیم روی گروه آبلی متناهی G یک گراف کیلی مکعبی همبند وجود دارد که تمام مقادیر ویژه ی ماتریس فاصله ی آن صحیح هستند اگر و تنها اگر G یکریخت با یکی از گروه های Z_4 ، Z_6 ، Z_4xZ_2 ، Z_6xZ_2  یا Z_2xZ_2xZ_2  باشد. علاوه بر این نشان می دهیم که، تحت یکریختی، تنها 5 گراف کیلی مکعبی همبند وجود دارد که تمام مقادیر ویژه ی ماتریس فاصله ی آن ها صحیح هستند.

    کلید واژگان: ماتریس فاصله، گراف کیلی، مقدار ویژه، نمایش تحویل ناپذیر
    Majid Arezoomand*
    Introduction

    In this paper, graphs are undirected and loop-free and groups are finite. By 𝐶𝑛, 𝐾𝑛 and 𝐾𝑚,𝑛 we mean the cycle graph with 𝑛 vertices, the complete graph with 𝑛 vertices and the complete bipartite graph with parts size 𝑚 and 𝑛, respectively. Also by 𝑍𝑛 and 𝑆𝑛, we mean the cyclic group of order 𝑛 and the symmetric group on 𝑛 symbols, respectively. Let Γ be a simple connected graph with vertex set {𝑣1 , … , 𝑣𝑛}. The distance between vertices 𝑣𝑖 and 𝑣𝑗 , denoted by 𝑑(𝑣𝑖 , 𝑣𝑗), is the length of a shortest path between them. The distance matrix of Γ, denoted by 𝐷Γ, is an 𝑛 × 𝑛 matrix whose (𝑖,𝑗)-entry is 𝑑(𝑣𝑖 , 𝑣𝑗). The distance characteristic polynomial of Γ, denoted by 𝜒𝐷(Γ) is det(𝜆𝐼 − 𝐷) and its zeros are the distance eigenvalues (in short 𝐷-eigenvalues) of Γ. If 𝜆 is a 𝐷-eigenvalue of Γ with multiplicity 𝑚, then we denote it by 𝜆 [𝑚] . Let 𝜆1 ≥ 𝜆2 ≥ ⋯ ≥ 𝜆𝑛 are the 𝐷-eigenvalues of Γ. Then 𝜆1 is called distance spectral radius of Γ and we denote it by 𝜌(Γ). Also the multiset {𝜆1 , … , 𝜆𝑛} is denoted by S𝑝𝑒𝑐𝐷(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let 𝐺 be a group and 𝑆 = 𝑆 −1 be a subset of 𝐺 not containing the identity element of 𝐺. The Cayley graph of 𝐺 with respect of 𝑆, denoted by C𝑎𝑦(𝐺, 𝑆), is a graph with vertex set 𝐺 and edge set {{𝑔, 𝑠𝑔}|𝑔 ∈ 𝐺, 𝑠 ∈ 𝑆}. C𝑎𝑦(𝐺, 𝑆) is a simple |𝑆|-regular graph. Let 𝑥, 𝑦 ∈ 𝐺. Then for all 𝑔 ∈ 𝐺, 𝑥 and 𝑦 are adjacent if and only if 𝑥𝑔 and 𝑦𝑔 are adjacent. This implies that 𝑑(𝑔, ℎ) = 𝑑(1, ℎ𝑔 −1 ) and 𝑑(𝑔) = 𝑑(1) for all 𝑔, ℎ ∈ 𝐺, where 𝑑(𝑥) = ∑𝑦∈𝐺 𝑑(𝑥, 𝑦). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, FosterGreenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ = C𝑎𝑦(𝐺, 𝑆) be a Cayley graph over a finite group 𝐺. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of 𝐺, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of 𝐺. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, 𝑛-prims, hexagonal torus network and cubic Cayley graphs over abelian groups.

    Results and discussion

    We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group 𝐺 admits a connected cubic distance integral Cayley graph if and only if 𝐺 is isomorphic to one of the groups 𝑍4 , 𝑍6 , 𝑍4 × 𝑍2 , 𝑍6 × 𝑍2 , or 𝑍2 × 𝑍2 × 𝑍2 . Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are 𝐾4 , 𝐾3,3 , 𝒫3 , 𝒫4 and 𝒫6 , where 𝒫𝑛 is the 𝑛-prism.

    Conclusion

    The following conclusions were drawn from this research.  The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G.  Exact formulas for 𝑛-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given.  Infinite family of distance integral Cayley graphs are constructed.  Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups.  One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.

    Keywords: Distance matrix, rreducible representation, Cayley graph, Eigenvalue
  • Somayeh Ahmadi, Ebrahim Vatandoost, Ali Bahraini

    Let $\Gamma=(V, E)$ be a simple graph. A set $C$ of vertices $\Gamma$ is an identifying set of $\Gamma$ if for every two vertices $x$ and $y$ the sets $N_{\Gamma}[x] \cap C$ and $N_{\Gamma}[y] \cap C$ are non-empty and different. Given a graph $\Gamma,$ the smallest size of an identifying set of $\Gamma$ is called the identifying code number of $\Gamma$ and is denoted by $\gamma^{ID}(\Gamma).$ Two vertices $x$ and $y$ are twins when $N_{\Gamma}[x]=N_{\Gamma}[y].$ Graphs with at least two twin vertices are not identifiable graph. In this paper, we study identifying code number of some Cayley graphs.

    Keywords: Domination, Identifying code, Cayley graph
  • مادلاین ال تحان، بیژن دواز*
    M. Al Tahan, B. Davvaz *

    The aim of this paper is to extend the notion of geometric groups to geometric hypergroups and to investigate the interaction between algebraic and geometric properties of hypergroups. In this regard, we first define a metric structure on hypergroups via word metric and present some examples on it by using generalized Cayley graphs over hypergroups. Then we study a large scale of geometry with respect to the structure of hypergroups and we prove that metric spaces of finitely generated hypergroups coming from different generating sets are quasi-isometric.

    Keywords: Cayley graph, hypergroup, geometric hypergroup
  • Tanakorn Udomworarat, Teerapong Suksumran*

    In this article, we study connections between components of the Cayley graph Cay(G,A), where A is an arbitrary subset of a group G, and cosets of the subgroup of G generated by A. In particular, we show how to construct generating sets of G if Cay(G,A) has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.

    Keywords: Cayley graph, Connected graph, Coset, Generating set, Graph component
  • M. N. Iradmusa*

    Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called emph{sharply $t$-transitive} if for any given pair of $t$-tuples, exactly one element of $S$ carries the first to the second. In addition, a subset $S$ of $S_n$ is called $t$-intersecting if $fix(h^{-1}g)geq t$ for any two distinct permutations $h$ and $g$ of $S$. In this paper, we prove that there are only two sharply $(n-2)$-transitive subsets of $S_n$ and finally we establish some relations between sharply $k$-transitive subsets and $t$-intersecting subsets of $S_n$ where $k,tin mathbb{Z}$ and $0leq tleq kleq n$.

    Keywords: Symmetric group, Sharply transitive set of permutations, Cayley graph, Intersecting set of permutations
  • Sh. Sedghi, D.-W. Lee*, N. Shobe

    Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its congraphs.

    Keywords: Common Neighborhood Graph, Cayley graph, Graph operation
  • F. Ramezani *
    ‎In this paper‎, ‎we investigate domination number‎, ‎$gamma$‎, ‎as well‎ ‎as signed domination number‎, ‎$gamma_{_S}$‎, ‎of all cubic Cayley‎ ‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎ ‎the domination and signed domination numbers of cubic graphs depend‎ on each other‎.
    Keywords: Cayley graph, ‎cubic graph‎, ‎cyclic group‎, ‎domination number‎, ‎signed domination number
  • Khadijeh Shamsi, Reza Ameri *, Saeed Mirvakili
    The purpose of this paper is the study of Cayley graph associated to a semihypergroup(or hypergroup). In this regards first  we associate a Cayley graph to every semihypergroup and then we study theproperties of this graph, such as  Hamiltonian cycles in this  graph.  Also, by some of examples we will illustrate  the properties and behavior of  these Cayley  graphs, in particulars we show that the properties of a Cayley graph associated to a semihypergroup is  completely   different  with respect to the Cayley graph associated to a  semigroup(group). Also, we briefly discuss on category of Cayley graphs associated to  semihypergroups and construct a functor from this category to the category of digraphs. Finally, we  give an application the  Cayley graph of a hypergroupoid to a social network.
    Keywords: Semihypergroup, Cayley graph, Fundamental relation, Category, Social network
  • M. Ghorbani *, A. Seyyed Hadi, F. Nowroozi

    A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

    Keywords: symmetric graph, Cayley graph, normal graph, arc-transitive graph
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