scholarly journals Recognizing Circulant Graphs of Prime Order in Polynomial Time

10.37236/1363 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Adjacency Matrix ◽  
Prime Order ◽  
Circulant Matrix ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Necessary Condition ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Coherent Configuration ◽  
Schur Ring

A circulant graph $G$ of order $n$ is a Cayley graph over the cyclic group ${\bf Z}_n.$ Equivalently, $G$ is circulant iff its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration ${\cal A}$ and, in particular, a Schur ring ${\cal S}$ isomorphic to ${\cal A}$. ${\cal A}$ can be associated without knowing $G$ to be circulant. If $n$ is prime, then by investigating the structure of ${\cal A}$ either we are able to find an appropriate ordering of the vertices proving that $G$ is circulant or we are able to prove that a certain necessary condition for $G$ being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in $n$.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
Author(s):  
J.W. Sander ◽  
T. Sander
Keyword(s):  
Adjacency Matrix ◽  
Prime Power ◽  
Prime Power Order ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

scholarly journals Pretty Good State Transfer on Circulant Graphs

10.37236/6388 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Hiranmoy Pal ◽  
Bikash Bhattacharjya
Keyword(s):  
Complex Number ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Edge Disjoint ◽  
State Transfer ◽  
Good State

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

scholarly journals The Geometric Kernel of Integral Circulant Graphs

10.37236/9764 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
J. W. Sander
Keyword(s):  
Adjacency Matrix ◽  
Euclidean Plane ◽  
Rotational Symmetry ◽  
Central Hole ◽  
Geometric Feature ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Notable Feature ◽  
Arithmetic Properties ◽  
Vertex Set

By a suitable representation in the Euclidean plane, each circulant graph $G$, i.e. a graph with a circulant adjacency matrix ${\mathcal A}(G)$, reveals its rotational symmetry and, as the drawing's most notable feature, a central hole, the so-called \emph{geometric kernel} of $G$. Every integral circulant graph $G$ on $n$ vertices, i.e. satisfying the additional property that all of the eigenvalues of ${\mathcal A}(G)$ are integral, is isomorphic to some graph $\mathrm{ICG}(n,\mathcal{D})$ having vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in \mathcal{D}\}$ for a uniquely determined set $\mathcal{D}$ of positive divisors of $n$. A lot of recent research has revolved around the interrelation between graph-theoretical, algebraic and arithmetic properties of such graphs. In this article we examine arithmetic implications imposed on $n$ by a geometric feature, namely the size of the geometric kernel of $\mathrm{ICG}(n,\mathcal{D})$.

scholarly journals Recognizing Circulant Graphs in Polynomial Time: An Application of Association Schemes

10.37236/1570 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Polynomial Time ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Circulant Graphs ◽  
Symmetry Properties ◽  
Coherent Configurations

In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism.

HAMILTONIAN PROPERTIES OF FAULTY RECURSIVE CIRCULANT GRAPHS

2002 ◽  
Vol 03 (03n04) ◽  
pp. 273-289 ◽  
Author(s):  
CHANG-HSIUNG TSAI ◽  
JIMMY J. M. TAN ◽  
YEN-CHU CHUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Path ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Regular Node ◽  
Hamiltonian Properties

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

scholarly journals Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

2019 ◽  
Vol 14 (2) ◽  
pp. 103-126
Author(s):  
József Borbély ◽  
András Sárközy
Keyword(s):  
Random Graphs ◽  
Adjacency Matrix ◽  
Binary Sequence ◽  
Binary Sequences ◽  
Circulant Graph ◽  
Large Families

AbstractIn the last decades many results have been proved on pseudo-randomness of binary sequences. In this series our goal is to show that using many of these results one can also construct large families of quasi-random, pseudo-random and strongly pseudo-random graphs. Indeed, it will be proved that if the first row of the adjacency matrix of a circulant graph forms a binary sequence which possesses certain pseudorandom properties (and there are many large families of binary sequences known with these properties), then the graph is quasi-random, pseudo-random or strongly pseudo-random, respectively. In particular, here in Part I we will construct large families of quasi-random graphs along these lines. (In Parts II and III we will present and study constructions for pseudo-random and strongly pseudo-random graphs, respectively.)

scholarly journals The integer {k}-domination number of circulant graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050055
Author(s):  
Yen-Jen Cheng ◽  
Hung-Lin Fu ◽  
Chia-An Liu
Keyword(s):  
Undirected Graph ◽  
Domination Number ◽  
Difference Set ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Minimum Value

Let [Formula: see text] be a simple undirected graph. [Formula: see text] is a circulant graph defined on [Formula: see text] with difference set [Formula: see text] provided two vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text]. For convenience, we use [Formula: see text] to denote such a circulant graph. A function [Formula: see text] is an integer [Formula: see text]-domination function if for each [Formula: see text], [Formula: see text] By considering all [Formula: see text]-domination functions [Formula: see text], the minimum value of [Formula: see text] is the [Formula: see text]-domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that if [Formula: see text], [Formula: see text], then the integer [Formula: see text]-domination number of [Formula: see text] is [Formula: see text].

scholarly journals An Interesting Property of a Class of Circulant Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Seyed Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Additive Group ◽  
Interesting Property ◽  
Circulant Graphs ◽  
Transitive Graph ◽  
Matrix Spectrum ◽  
Adjacency Matrix Spectrum

Suppose thatΠ=Cay(Zn,Ω)andΛ=Cay(Zn,Ψm)are two Cayley graphs on the cyclic additive groupZn, wherenis an even integer,m=n/2+1,Ω=t∈Zn∣t  is  odd, andΨm=Ω∪{n/2}are the inverse-closed subsets ofZn-0. In this paper, it is shown thatΠis a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum ofΠ. Finally, we show that ifn≥8andn/2is an even integer, then the adjacency matrix spectrum ofΛisn/2+11,1-n/21,1n-4/2,-1n/2(we write multiplicities as exponents).

scholarly journals On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

2020 ◽  
Vol 27 (01) ◽  
pp. 87-94
Author(s):  
A.D. Mednykh ◽  
I.A. Mednykh
Keyword(s):  
Rational Function ◽  
Generating Function ◽  
Spanning Trees ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Integer Coefficients ◽  
Number Formula ◽  
Number Of Spanning Trees

Let [Formula: see text] be the generating function for the number [Formula: see text] of spanning trees in the circulant graph Cn(s1, s2, …, sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, …, sk, n) of odd valency. We illustrate the obtained results by a series of examples.

Integral circulant graphs with four distinct eigenvalues

2018 ◽  
Vol 10 (05) ◽  
pp. 1850057 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Raja
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Finite Cyclic Group ◽  
Vertex Set

Let [Formula: see text], the finite cyclic group of order [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. The circulant graph [Formula: see text] is the undirected graph having the vertex set [Formula: see text] and edge set [Formula: see text]. Let [Formula: see text] be a set of positive, proper divisors of the integer [Formula: see text]. In this paper, by using [Formula: see text] we characterize certain connected integral circulant graphs with four distinct eigenvalues.

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