scholarly journals On Automorphisms of the Double Cover of a Circulant Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ademir Hujdurović ◽  
Đorđe Mitrović ◽  
Dave Witte Morris
Keyword(s):  
Prime Number ◽  
Direct Product ◽  
Double Cover ◽  
Circulant Graph ◽  
Circulant Graphs

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of $X$ have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

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.

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 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 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 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.

scholarly journals PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS

2007 ◽  
Vol 05 (03) ◽  
pp. 417-430 ◽  
Author(s):  
NITIN SAXENA ◽  
SIMONE SEVERINI ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Quantum Dynamics ◽  
Perfect State ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Odd Order ◽  
State Transfer ◽  
Network Topologies ◽  
Perfect State Transfer ◽  
Exact Bounds

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals On hamiltonian decompositions of tensor products of graphs

2019 ◽  
Vol 13 (1) ◽  
pp. 178-202
Author(s):  
P. Paulraja ◽  
Kumar Sampath
Keyword(s):  
Graph Theory ◽  
Tensor Product ◽  
Tensor Products ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Hamiltonian Decompositions ◽  
Hamiltonian Decomposition

Finding a hamiltonian decomposition of G is one of the challenging problems in graph theory. We do not know for what classes of graphs G and H, their tensor product G x H is hamiltonian decomposable. In this paper, we have proved that, if G is a hamiltonian decomposable circulant graph with certain properties and H is a hamiltonian decomposable multigraph, then G x H is hamiltonian decomposable. In particular, tensor products of certain sparse hamiltonian decomposable circulant graphs are hamiltonian decomposable.

scholarly journals NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

2010 ◽  
Vol 83 (1) ◽  
pp. 87-95
Author(s):  
KA HIN LEUNG ◽  
VINH NGUYEN ◽  
WASIN SO
Keyword(s):  
Regular Graph ◽  
Elementary Proof ◽  
Explicit Construction ◽  
Simple Graph ◽  
Regular Graphs ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Image Position ◽  
Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

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