scholarly journals 2-Restricted Edge Connectivity of Wheel Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Wei Feng ◽  
Shiying Wang
Keyword(s):  
Fault Tolerance ◽  
Cayley Graph ◽  
Connected Graph ◽  
Large Scale ◽  
Edge Connectivity ◽  
Simple Connected Graph

The g -restricted edge connectivity ( g -REC) is an efficient index in evaluating the reliability and fault tolerance of large-scale processing systems. Assume that G = V , E is a simple connected graph and F ⊆ E . The g -REC of G , denoted by λ g G , is the minimum F such that G − F is disconnected and any component has at least g nodes. The n -dimensional wheel network CW n , a kind of Cayley graph, possesses many desired features. In this paper, we establish that λ 2 CW n is 4 n − 6 for n ≥ 5 , and CW n is λ 2 -optimal.

The Laplacian quadratic form and edge connectivity of a graph

2015 ◽  
Vol 30 ◽  
Author(s):  
William Watkins
Keyword(s):  
Quadratic Form ◽  
Connected Graph ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
Edge Connectivity ◽  
Integral Quadratic Form ◽  
Simple Connected Graph

Let G be a simple connected graph with associated positive semidefinite integral quadratic form Q(x) = \sum (x(i) − x(j))^2, where the sum is taken over all edges ij of G. It is showed that the minimum positive value of Q(x) for x ∈ Z_n equals the edge connectivity of G. By restricting Q(x) to x ∈ Z_{n−1} × {0}, the quadratic form becomes positive definite. It is also showed that the number of minimal disconnecting sets of edges of G equals twice the number of vectors x ∈ Z_{n−1} ×{0} for which the form Q attains its minimum positive value.

Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

2021 ◽  
pp. 1-26
Author(s):  
Yihong Wang ◽  
Cheng-Kuan Lin ◽  
Shuming Zhou ◽  
Tao Tian
Keyword(s):  
Big Data ◽  
Fault Tolerance ◽  
Connected Graph ◽  
Interconnection Networks ◽  
Large Scale ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Multiprocessor Systems ◽  
Edge Disjoint ◽  
Cube Formula

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph [Formula: see text] is called strong Menger (edge) connected if, for any two distinct vertices [Formula: see text] and [Formula: see text], there are [Formula: see text] vertex (edge)-disjoint paths between them. Exchanged hypercube [Formula: see text], as a variant of hypercube [Formula: see text], remains lots of preferable fault tolerant properties of hypercube. In this paper, we show that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are strong Menger (edge) connected, respectively. Moreover, as a by-product, for dual cube [Formula: see text], one popular generalization of hypercube, [Formula: see text] is also showed to be strong Menger (edge) connected, where [Formula: see text].

The inverse sum indeg index of graphs with some given parameters

2018 ◽  
Vol 10 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Hanlin Chen ◽  
Hanyuan Deng
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Extremal Graphs ◽  
Edge Connectivity ◽  
Degree Of A Vertex ◽  
Graph Parameters ◽  
Simple Connected Graph

Let [Formula: see text] be a simple connected graph. The inverse sum indeg index of [Formula: see text], denoted by [Formula: see text], is defined as the sum of the weights [Formula: see text] of all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex in [Formula: see text]. In this paper, we derive some bounds for the inverse sum indeg index in terms of some graph parameters, such as vertex (edge) connectivity, chromatic number, vertex bipartiteness, etc. The corresponding extremal graphs are characterized, respectively.

A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs

2019 ◽  
Vol 63 (9) ◽  
pp. 1372-1384
Author(s):  
Zuwen Luo ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Connected Graph ◽  
Cayley Graphs ◽  
Network Structures ◽  
Vertex Connectivity ◽  
Wheel Graphs

Abstract Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$-vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$. The cardinality of a minimum $R^{k}$-vertex-cut is the $R^{k}$-vertex-connectivity of $G$ and is denoted by $\kappa ^{k}(G)$. $R^{k}$-vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$-vertex-connectivity and $R^{2}$-vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $\kappa ^{1}(AW_{n})=4n-7$ for $n\geq 6$; $\kappa ^{2}(AW_{n})=6n-12$ for $n\geq 6$.

Research and Design of the Distributed Mass Small File Storage System Based on WCF

2013 ◽  
Vol 765-767 ◽  
pp. 1087-1091
Author(s):  
Hong Lin ◽  
Shou Gang Chen ◽  
Bao Hui Wang
Keyword(s):  
Fault Tolerance ◽  
Distributed Computing ◽  
Data Storage ◽  
Large Scale ◽  
Storage System ◽  
Hardware Platform ◽  
File Storage ◽  
Computing Framework ◽  
Research And Design ◽  
Small File

Recently, with the development of Internet and the coming of new application modes, data storage has some new characters and new requirements. In this paper, a Distributed Computing Framework Mass Small File storage System (For short:Dnet FS) based on Windows Communication Foundation in .Net platform is presented, which is lightweight, good-expansibility, running in cheap hardware platform, supporting Large-scale concurrent access, and having certain fault-tolerance. The framework of this system is analyzed and the performance of this system is tested and compared. All of these prove this system meet requirements.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

2007 ◽  
Vol 3 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

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