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.

scholarly journals Some Properties of Unitary Cayley Graphs

10.37236/963 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Cayley Graph ◽  
Chromatic Number ◽  
Cayley Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Vertex Connectivity ◽  
Euler Function ◽  
Vertex Set ◽  
Unitary Cayley Graph

The unitary Cayley graph $X_n$ has vertex set $Z_n=\{0,1, \ldots ,n-1\}$. Vertices $a, b$ are adjacent, if gcd$(a-b,n)=1$. For $X_n$ the chromatic number, the clique number, the independence number, the diameter and the vertex connectivity are determined. We decide on the perfectness of $X_n$ and show that all nonzero eigenvalues of $X_n$ are integers dividing the value $\varphi(n)$ of the Euler function.

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.

Clique Number and Girth of the Rough Ideal-Based Rough Edge Cayley Graph

2019 ◽  
Vol 15 (03) ◽  
pp. 479-487
Author(s):  
B. Praba ◽  
X. A. Benazir Obilia
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Clique Number

In this paper, clique number and girth of the rough ideal-based rough edge Cayley graphs [Formula: see text] and [Formula: see text], where [Formula: see text] is the rough Ideal of the rough semiring [Formula: see text] and [Formula: see text] contains the nontrivial elements of [Formula: see text], are evaluated. We prove that the clique number of [Formula: see text] is [Formula: see text] and the clique number of [Formula: see text] is [Formula: see text]. Girth of [Formula: see text] and [Formula: see text] is 3. These concepts are illustrated with examples.

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 On the Unitary Cayley Graph of a Finite Ring

10.37236/206 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Reza Akhtar ◽  
Megan Boggess ◽  
Tiffany Jackson-Henderson ◽  
Isidora Jiménez ◽  
Rachel Karpman ◽  
...  
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Finite Ring ◽  
Perfect Graphs ◽  
Edge Connectivity ◽  
Unitary Cayley Graph

We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.

scholarly journals Hardness of computing clique number and chromatic number for Cayley graphs

2017 ◽  
Vol 62 ◽  
pp. 147-166
Author(s):  
Chris Godsil ◽  
Brendan Rooney
Keyword(s):  
Chromatic Number ◽  
Cayley Graphs ◽  
Clique Number

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 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 Gutman Index and Harary Index of Unitary Cayley Graphs

2018 ◽  
Vol 7 (3) ◽  
pp. 1243 ◽  
Author(s):  
Roshan Philipose ◽  
Sarasija P B
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Harary Index ◽  
Vertex Set ◽  
Gutman Index ◽  
Unitary Cayley Graph

In this paper, we determine the Gutman Index and Harary Index of Unitary Cayley Graphs. The Unitary Cayley Graph Xn is the graph with vertex set  V(Xn) ={u|u∈ Zn} and edge set {uv|gcd(u−v, n) = 1 and u, v ∈ Zn }, where Zn ={0,1,...,n−1}.

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