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.

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.

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 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 The hyperbolicity constant of infinite circulant graphs

2017 ◽  
Vol 15 (1) ◽  
pp. 800-814
Author(s):  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Large Class ◽  
Cyclic Group ◽  
Metric Space ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Geodesic Triangle ◽  
Group Of Automorphisms ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Hyperbolicity Constant

Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant 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 L(3,1)-labeling of circulant graphs

2021 ◽  
pp. 2150093
Author(s):  
Soumya Bhoumik ◽  
Sarbari Mitra
Keyword(s):  
Cayley Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
The Difference

An [Formula: see text]-labeling of a graph [Formula: see text] is an assignment of non-negative integers to the vertices such that if two vertices [Formula: see text] and [Formula: see text] are adjacent then they receive labels that differ by at least [Formula: see text], and when [Formula: see text] and [Formula: see text] are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least one. The span [Formula: see text] of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let [Formula: see text] denote the least [Formula: see text] such that [Formula: see text] admits an [Formula: see text]-labeling using labels from [Formula: see text]. A Cayley graph of group [Formula: see text] is called circulant graph of order [Formula: see text], if [Formula: see text]. In this paper initially we investigate the [Formula: see text]-labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of [Formula: see text]-labeling for any circulants of order [Formula: see text].

scholarly journals On Embeddings of Circulant Graphs

10.37236/4013 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Marston Conder ◽  
Ricardo Grande
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Circulant Graphs ◽  
Transitive Action ◽  
Planar Case ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Orientable Surfaces

A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).

scholarly journals Frobenius groups of low rank

2021 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Frobenius Group ◽  
Low Rank ◽  
Special Situation ◽  
Frobenius Groups ◽  
Frobenius Kernel

AbstractLet G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

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.

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