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.

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 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.

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}.

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.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

scholarly journals Random spanning trees of Cayley graphs and an associated compactification of semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 611-620
Author(s):  
Steven N. Evans
Keyword(s):  
Random Walk ◽  
Cayley Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finite Tree ◽  
Generating Set ◽  
Countably Infinite

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

Cayley graphs of graded rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850116
Author(s):  
Saadoun Mahmoudi ◽  
Shahram Mehry ◽  
Reza Safakish
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Graded Rings ◽  
Graded Ring ◽  
Total Graph ◽  
Hamming Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Cayley Sum Graph

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

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.

Automorphisms of a Family of Cubic Graphs

2013 ◽  
Vol 20 (03) ◽  
pp. 495-506 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Mohsen Ghasemi
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

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