The number of shortest paths in the (n, k)-star graph

2014 ◽  
Vol 06 (04) ◽  
pp. 1450051 ◽  
Author(s):  
Eddie Cheng ◽  
Ke Qiu ◽  
Zhizhang Shen
Keyword(s):  
Shortest Paths ◽  
The Other ◽  
Alternating Group ◽  
Star Graph ◽  
Arrangement Graph

We count the shortest paths, not necessarily disjoint, between any two vertices in an (n, k)-star graph by counting the minimum factorizations of a permutation in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other structures closely related to the (n, k)-star graph, such as the alternating group networks. Furthermore, the techniques made use of in this paper can be applied to solve the same problem for some of the other structures such as the arrangement graph.

THE Qn,k,m GRAPH: A COMMON GENERALIZATION OF VARIOUS POPULAR INTERCONNECTION NETWORKS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350011 ◽  
Author(s):  
EDDIE CHENG ◽  
NART SHAWASH
Keyword(s):  
Interconnection Networks ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Star Graph ◽  
Strong Connectivity ◽  
Common Generalization ◽  
Arrangement Graph ◽  
Special Case ◽  
Alternating Group Graph

The star graph and the alternating group graph were introduced as competitive alternatives to the hypercube, and they are indeed superior over the hypercube under many measures. However, they do suffer from scaling issues. To address this, different generalizations, namely, the (n,k)-star graph and the arrangement graph were introduced to address this shortcoming. From another direction, the star graph was recognized as a special case of Cayley graphs whose generators can be associated with a tree. Nevertheless, all these networks appear to be very different and yet share many properties. In this paper, we will solve this mystery by providing a common generalization of all these networks. Moreover, we will show that these networks have strong connectivity properties.

On permutability graphs of subgroups of groups

2015 ◽  
Vol 07 (02) ◽  
pp. 1550012 ◽  
Author(s):  
R. Rajkumar ◽  
P. Devi
Keyword(s):  
Finite Groups ◽  
The Other ◽  
Star Graph ◽  
Infinite Groups ◽  
Vertex Set

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C3-free, C5-free, K4-free, K5-free, K1,4-free, K2,3-free or Pn-free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q8, S3 and A4 by using their permutability graphs of subgroups.

A Note of Independent Number and Domination Number of Qn,k,m-Graph

2019 ◽  
Vol 29 (03) ◽  
pp. 1950011
Author(s):  
Jiafei Liu ◽  
Shuming Zhou ◽  
Zhendong Gu ◽  
Yihong Wang ◽  
Qianru Zhou
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Domination Number ◽  
Multiprocessor System ◽  
Star Graph ◽  
Minimum Cardinality ◽  
Arrangement Graph ◽  
Independent Number ◽  
Potential Applications ◽  
By Products

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

scholarly journals Biomarker Identification for Prostate Cancer and Lymph Node Metastasis from Microarray Data and Protein Interaction Network Using Gene Prioritization Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Carlos Roberto Arias ◽  
Hsiang-Yuan Yeh ◽  
Von-Wun Soo
Keyword(s):  
Prostate Cancer ◽  
Protein Interaction ◽  
Microarray Data ◽  
Shortest Paths ◽  
Gene Prioritization ◽  
The Other ◽  
Biomarker Identification ◽  
Lymph Nodes Metastasis ◽  
Prioritization Method ◽  
Voting Scheme

Finding a genetic disease-related gene is not a trivial task. Therefore, computational methods are needed to present clues to the biomedical community to explore genes that are more likely to be related to a specific disease as biomarker. We present biomarker identification problem using gene prioritization method called gene prioritization from microarray data based on shortest paths, extended with structural and biological properties and edge flux using voting scheme (GP-MIDAS-VXEF). The method is based on finding relevant interactions on protein interaction networks, then scoring the genes using shortest paths and topological analysis, integrating the results using a voting scheme and a biological boosting. We applied two experiments, one is prostate primary and normal samples and the other is prostate primary tumor with and without lymph nodes metastasis. We used 137 truly prostate cancer genes as benchmark. In the first experiment, GP-MIDAS-VXEF outperforms all the other state-of-the-art methods in the benchmark by retrieving the truest related genes from the candidate set in the top 50 scores found. We applied the same technique to infer the significant biomarkers in prostate cancer with lymph nodes metastasis which is not established well.

TOPOLOGICAL PROPERTIES OF THE (n,k)-STAR GRAPH

1998 ◽  
Vol 09 (02) ◽  
pp. 235-248 ◽  
Author(s):  
WEI-KUO CHIANG ◽  
RONG-JAYE CHEN
Keyword(s):  
Average Distance ◽  
Point Of View ◽  
Attractive Alternative ◽  
Star Graph ◽  
Topological Properties ◽  
Major Drawback ◽  
Arrangement Graph ◽  
Different Types ◽  
Theory Point ◽  
Very High

The star graph, though an attractive alternative to the hypercube, has a major drawback in that the number of nodes for an n-star graph must be n!, and thus considerably limits the choice of the number of nodes in the graph. In order to alleviate this drawback, the arrangement graph was recently proposed as a generalization of the star graph topology. The arrangement graph provides more flexibility than the star graph in choosing the number of nodes, but the degree of the resulting network may be very high. To overcome that disadvantage, this paper presents another generalization of the star graph, called the (n,k)-star graph. We examine some topological properties of the (n,k)-star graph from the graph-theory point of view. It is shown that two different types of edges in the (n,k)-star prevent it from being edge-symmetric, but edges in each class are essentially symmetric with respect to each other. Also, the diameter and the exact average distance of the (n,k)-star graph are derived. In addition, the fault-diameter for the (n,k)-star graph is shown to be at most the fault-free diameter plus three.

scholarly journals OD-characterization of alternating groups Ap+d

2017 ◽  
Vol 15 (1) ◽  
pp. 1090-1098
Author(s):  
Yong Yang ◽  
Shitian Liu ◽  
Zhanghua Zhang
Keyword(s):  
The Other ◽  
Alternating Group ◽  
Alternating Groups ◽  
Other Hand

Abstract Let An be an alternating group of degree n. Some authors have proved that A10, A147 and A189 cannot be OD-characterizable. On the other hand, others have shown that A16, A23+4, and A23+5 are OD-characterizable. We will prove that the alternating groups Ap+d except A10, are OD-characterizable, where p is a prime and d is a prime or equals to 4. This result generalizes other results.

Natural constructions of the Mathieu groups

1989 ◽  
Vol 106 (3) ◽  
pp. 423-429 ◽  
Author(s):  
R. T. Curtis
Keyword(s):  
Symmetric Groups ◽  
The Other ◽  
Simple Groups ◽  
Alternating Group ◽  
Transitive Permutation Group ◽  
French Mathematician ◽  
Exceptional Groups ◽  
Transitive Groups ◽  
Mathieu Groups ◽  
Alternating And Symmetric Groups

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

Subgraph Reliability of Alternating Group Graph With Uniform and Nonuniform Vertex Fault-Free Probabilities

2020 ◽  
Author(s):  
Yanze Huang ◽  
Limei Lin ◽  
Li Xu
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fault Model ◽  
Alternating Group ◽  
Multiprocessor System ◽  
Single Vertex ◽  
Arrangement Graph ◽  
Uniform Probability ◽  
Better Than ◽  
Alternating Group Graph

Abstract As the size of a multiprocessor system grows, the probability that faults occur in this system increases. One measure of the reliability of a multiprocessor system is the probability that a fault-free subsystem of a certain size still exists with the presence of individual faults. In this paper, we use the probabilistic fault model to establish the subgraph reliability for $AG_n$, the $n$-dimensional alternating group graph. More precisely, we first analyze the probability $R_n^{n-1}(p)$ that at least one subgraph with dimension $n-1$ is fault-free in $AG_n$, when given a uniform probability of a single vertex being fault-free. Since subgraphs of $AG_n$ intersect in rather complicated manners, we resort to the principle of inclusion–exclusion by considering intersections of up to five subgraphs and obtain an upper bound of the probability. Then we consider the probabilistic fault model when the probability of a single vertex being fault-free is nonuniform, and we show that the upper bound under these two models is very close to the lower bound obtained in a previous result, and it is better than the upper bound deduced from that of the arrangement graph, which means that the upper bound we obtained is very tight.

The number of shortest paths in the arrangement graph

2013 ◽  
Vol 240 ◽  
pp. 191-204 ◽  
Author(s):  
Eddie Cheng ◽  
Jerrold W. Grossman ◽  
Ke Qiu ◽  
Zhizhang Shen
Keyword(s):  
Shortest Paths ◽  
Arrangement Graph

scholarly journals Pelabelan Selimut Bintang Ajaib Super Pada Graf Bintang

2021 ◽  
Vol 18 (1) ◽  
pp. 95-109
Author(s):  
N Mattiro ◽  
I W Sudarsana
Keyword(s):  
Research Methodology ◽  
Simple Graph ◽  
The Other ◽  
Star Graph ◽  
Literature Study ◽  
Covering Graph ◽  
Edge Covering ◽  
The Given

Let  be a simple graph. An edge covering of  is a family of subgraphs  such that each edge of graph  belongs to at least one of the ,  subgraphs. If each  is isomorphic with the given graph , then it is said that contains a  covering. The graph G contains a covering  and   the bijectif function  is said an the magic labeling of a graph G if for each subgraph  of  is isomorphic to , so that is a constant. It is said that the graph G has a super magic if  in this case, the graph G which can be labeled with  magic is called the covering graph  magic. A star graph with n points is a graph with  points and  sides, where point is  degree and the other  point has degree  denoted by . This study aims to determine the presence of covering labeling for the super-magic star on the  star graph. The research methodology is literature study. The results show that the  star graph for   has   magic covering labeling with magic constants for all covering is  and the super-magic covering labeling with magic constants for all covering is .

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