scholarly journals Hamiltonian properties on a class of circulant interconnection networks

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 71-85 ◽  
Author(s):  
Milan Basic
Keyword(s):  
Cayley Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Quantum Spin ◽  
Perfect State ◽  
Circulant Graphs ◽  
Hamiltonian Paths ◽  
Unitary Cayley Graph

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

scholarly journals On the Automorphism Group of Integral Circulant Graphs

10.37236/555 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Milan Bašić ◽  
Aleksandar Ilić
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Quantum Spin ◽  
Perfect State ◽  
Circulant Graphs ◽  
Spin Networks ◽  
Chemical Graph Theory ◽  
State Transfer ◽  
Vertex Set ◽  
Unitary Cayley Graph

The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0, 1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a set of divisors of $n$. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph $X_n (1)$ and the disconnected graph $X_n (d)$. In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1, p^k))$ for $n$ being a prime power.

scholarly journals Borel Cayley Graph-Based Topology Control for Consensus Protocol in Wireless Sensor Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Junghun Ryu ◽  
Jaewook Yu ◽  
Eric Noel ◽  
K. Wendy Tang
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Interconnection Networks ◽  
Topology Control ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Wireless Sensor ◽  
Control Algorithms ◽  
Transmission Range ◽  
Consensus Protocol

Borel Cayley graphs have been shown to be an efficient candidate topology in interconnection networks due to their small diameter, short path length, and low degree. In this paper, we propose topology control algorithms based on Borel Cayley graphs. In particular, we propose two methods to assign node IDs of Borel Cayley graphs as logical topologies in wireless sensor networks. The first one aims at minimizing communication distance between nodes, while the entire graph is imposed as a logical topology; while the second one aims at maximizing the number of edges of the graph to be used, while the network nodes are constrained with a finite radio transmission range. In the latter case, due to the finite transmission range, the resultant topology is an “incomplete” version of the original BCG. In both cases, we apply our algorithms in consensus protocol and compare its performance with that of the random node ID assignment and other existing topology control algorithms. Our simulation indicates that the proposed ID assignments have better performance when consensus protocols are used as a benchmark application.

On the Diameter of Unitary Cayley Graphs of Rings

2016 ◽  
Vol 59 (3) ◽  
pp. 652-660
Author(s):  
Huadong Su
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Unitary Cayley Graph

AbstractThe unitary Cayley graph of a ringR, denoted Γ(R), is the simple graph defined on all elements ofR, and where two verticesxandyare adjacent if and only ifx−yis a unit inR. The largest distance between all pairs of vertices of a graphGis called the diameter ofGand is denoted by diam(G). It is proved that for each integern≥ 1, there exists a ringRsuch that diam(Γ(R)) =n. We also show that diam(Γ(R)) ∊ {1, 2, 3,∞} for a ringRwithR/J(R) self-injective and classify all those rings with diam(Γ(R)) = 1, 2, 3, and ∞, respectively.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

scholarly journals On the Energy of Unitary Cayley Graphs

10.37236/262 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
H. N. Ramaswamy ◽  
C. R. Veena
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Prime Divisors ◽  
Unitary Cayley Graph

In this note we obtain the energy of unitary Cayley graph $X_{n}$ which extends a result of R. Balakrishnan for power of a prime and also determine when they are hyperenergetic. We also prove that ${E(X_{n})\over 2(n-1)}\geq{2^{k}\over 4k}$, where $k$ is the number of distinct prime divisors of $n$. Thus the ratio ${E(X_{n})\over 2(n-1)}$, measuring the degree of hyperenergeticity of $X_{n}$, grows exponentially with $k$.

scholarly journals UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)

2021 ◽  
Vol 7 (2) ◽  
pp. 43
Author(s):  
Reza Jahani-Nezhad ◽  
Ali Bahrami
Keyword(s):  
Cayley Graph ◽  
Chromatic Number ◽  
Cayley Graphs ◽  
Clique Number ◽  
Unit Graph ◽  
Unitary Cayley Graph

Let \({E}_{n}\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}_{n})\) and \(G_{{E}_{n}}\), the unit graph and the unitary Cayley graph of \({E}_{n}\), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(E_{n})\) and \(G_{E_{n}}\) are Hamiltonian.

Fault-Tolerant Maximal Local-Connectivity on Cayley Graphs Generated by Transpositions

2019 ◽  
Vol 30 (08) ◽  
pp. 1301-1315 ◽  
Author(s):  
Liqiong Xu ◽  
Shuming Zhou ◽  
Weihua Yang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Local Connectivity ◽  
Communication Links ◽  
Restricted Condition ◽  
Vertex Disjoint

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.

scholarly journals FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR

2008 ◽  
Vol 85 (2) ◽  
pp. 269-282 ◽  
Author(s):  
ALISON THOMSON ◽  
SANMING ZHOU
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Interconnection Networks ◽  
Frobenius Group ◽  
Optimal Routing ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Frobenius Kernel ◽  
Even Order

AbstractA first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group $K \rtimes H$ such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

scholarly journals Polynomials of unitary Cayley graphs

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2079-2086
Author(s):  
Milan Basic ◽  
Aleksandar Ilic
Keyword(s):  
Cayley Graph ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Minimal Spanning Trees

The unitary Cayley graph Xn has the vertex set Zn = {0,1,2,..., n-1} and vertices a and b are adjacent, if and only if gcd(a-b,n) = 1. In this paper, we present some properties of the clique, independence and distance polynomials of the unitary Cayley graphs and generalize some of the results from [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. In addition, using some properties of Laplacian polynomial we determine the number of minimal spanning trees of any unitary Cayley graph.

scholarly journals A Note on Quadrangular Embedding of Abelian Cayley Graphs

2016 ◽  
Vol 17 (3) ◽  
pp. 331 ◽  
Author(s):  
João Eloir Strapasson ◽  
Sueli Irene Rodrigues Costa ◽  
Marcelo Muniz
Keyword(s):  
Lower Bound ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Cartesian Product ◽  
Circulant Graphs ◽  
Special Cases

The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.  

Sign in / Sign up

Export Citation Format

Share Document