scholarly journals On 4-valent Frobenius circulant graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Sanming Zhou
Keyword(s):  
Number Theory ◽  
Interconnection Networks ◽  
Prime Factor ◽  
Additive Group ◽  
Sense Making ◽  
Recursive Construction ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
International Audience ◽  
Congruence Equation

Graph Theory International audience A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.

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.

CONSTRUCTING MULTIPLE INDEPENDENT SPANNING TREES ON RECURSIVE CIRCULANT GRAPHS G(2m, 2)

2010 ◽  
Vol 21 (01) ◽  
pp. 73-90 ◽  
Author(s):  
JINN-SHYONG YANG ◽  
JOU-MING CHANG ◽  
SHYUE-MING TANG ◽  
YUE-LI WANG
Keyword(s):  
Interconnection Networks ◽  
Spanning Trees ◽  
Fault Tolerant ◽  
Network Communication ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Recursive Structure ◽  
Computing Systems ◽  
Independent Spanning Trees ◽  
Optimal Construction

A recursive circulant graph G(N,d) has N = cdm vertices labeled from 0 to N - 1, where d ⩾ 2, m ⩾ 1, and 1 ⩽ c < d, and two vertices x,y ∈ G(N,d) are adjacent if and only if there is an integer k with 0 ⩽ k ⩽ ⌈ log d N⌉ - 1 such that x ± dk ≡ y ( mod N). With the aid of recursive structure, such class of graphs has many attractive features and was considered as a topology of interconnection networks for computing systems. The design of multiple independent spanning trees (ISTs) has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. In the previous work of Yang et al. (2009), we provided a constructing scheme to build k ISTs on G(cdm,d) with d ⩾ 3, where k is the connectivity of G(cdm,d). However, the proposed constructing rules cannot be applied to the case of d = 2. For the integrity of solving the IST problem on recursive circulant graphs, this paper deals with the case of G(2m,2) using a set of different constructing rules. Especially, we show that the heights of ISTs for G(2m,2) are lower than the known optimal construction of hypercubes with the same number of vertices.

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 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 FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

2019 ◽  
pp. 781
Author(s):  
Narjes Seyedi ◽  
Hamid Reza Maimani
Keyword(s):  
Fault Tolerant ◽  
Additive Group ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace $ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right)  \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$.

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.

Some comments on inverse arithmetic functions

2004 ◽  
Vol 89 (516) ◽  
pp. 403-408
Author(s):  
P. G. Brown
Keyword(s):  
Number Theory ◽  
Prime Factor ◽  
Arithmetic Functions ◽  
Natural Numbers ◽  
Finite Mathematics ◽  
Factor Decomposition

In many of the basic courses in Number Theory, Finite Mathematics and Cryptography we come across the so-called arithmetic functions such as ϕn), σ(n), τ(n), μ(n), etc, whose domain is the set of natural numbers. These functions are well known and evaluated through the prime factor decomposition of n. It is less well known that these functions possess inverses (with respect to Dirichlet multiplication) which have interesting properties and applications.

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].

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