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 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.

scholarly journals On the independent spanning trees of recursive circulant graphs G(cdm,d) with d>2

2009 ◽  
Vol 410 (21-23) ◽  
pp. 2001-2010 ◽  
Author(s):  
Jinn-Shyong Yang ◽  
Jou-Ming Chang ◽  
Shyue-Ming Tang ◽  
Yue-Li Wang
Keyword(s):  
Spanning Trees ◽  
Circulant Graphs ◽  
Independent Spanning Trees

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 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 An Efficient Exchanged Hyper Cube for Parallel and Distributed Network

2019 ◽  
Vol 8 (2S10) ◽  
pp. 821-829
Keyword(s):  
Interconnection Networks ◽  
Distributed Network ◽  
Recursive Structure ◽  
Distributed Computing Systems ◽  
Computing Systems ◽  
Processing Power ◽  
Communication Time ◽  
Hypercube Topology ◽  
Tolerance Simple ◽  
Parallel Distribution

Enormously parallel distribution memory designs are accepting and expanding regard to satisfy the expanding need on processing power. Numerous topologies have been projected for interconnecting the processors of distributed computing systems. The hypercube topology has attracted significant consideration because of a significant number of attractive properties. The engaging properties of the hypercube topology, for example, vertex and edge balance, recursive structure, logarithmic diameter, maximally fault-tolerance, simple routing and broadcasting, and the capacity to recreate other interconnection systems with least overhead have made it a brilliant possibility for some parallel processing applications. Numerous varieties of the hypercube topology have been accounted for the literature, mostly to add the computational power of the hypercube. One of the gorgeous versions of the hypercube was introduced for the improvement of the presented Exchanged hypercube. An Exchanged hypercube has the equivalent structural complexities of the hypercube. It protects the gorgeous properties of the hypercube and diameter the communication time by dropping the diameter by a factor of two. This paper presents the fundamental communication and some of the essential operations normally required in parallel computing on the Exchanged hypercube interconnection networks.

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 Complexity of discrete Seifert foliations over a graph

2019 ◽  
Vol 486 (4) ◽  
pp. 411-415
Author(s):  
Young Soo Kwon ◽  
A. D. Mednykh ◽  
I. A. Mednykh
Keyword(s):  
Chebyshev Polynomials ◽  
Spanning Trees ◽  
Infinite Family ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Generalized Petersen Graphs ◽  
The Family ◽  
Petersen Graphs ◽  
Discrete Torus

In the present paper, we study the complexity of an infinite family of graphs Hn = Hn(G1, G2, ..., Gm) that are discrete Seifert foliations over a graph H on m vertices with fibers G1, G2, ..., Gm. Each fiber Gi = Cn(si,1, si,2, ..., si,ki) of this foliation is the circulant graph on n vertices with jumps si,1, si,2, ..., si,ki. The family of discrete Seifert foliations is sufficiently large. It includes the generalized Petersen graphs, I-graphs, Y-graphs, H-graphs, sandwiches of circulant graphs, discrete torus graph and others. We obtain a closed formula for the number t(n) of spanning trees in Hn in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as n → ∞.

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