Note on Reliability Analysis of Cartesian Product of Networks for Components

2021 ◽  
pp. 2142010
Author(s):  
Litao Guo ◽  
Jun Ge
Keyword(s):  
Reliability Analysis ◽  
Critical Parameter ◽  
Cartesian Product ◽  
Schwarz Inequality ◽  
Large Network ◽  
Vertex Number ◽  
Cartesian Product Of Graphs ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Text Component

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

scholarly journals γ-Graphs of Trees

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 153 ◽  
Author(s):  
Stephen Finbow ◽  
Christopher M. van Bommel
Keyword(s):  
Cartesian Product ◽  
Dominating Sets ◽  
Single Vertex ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Vertex Set ◽  
Cartesian Product Graphs ◽  
Product Of Graphs

For a graph G = ( V , E ) , the γ -graph of G, denoted G ( γ ) = ( V ( γ ) , E ( γ ) ) , is the graph whose vertex set is the collection of minimum dominating sets, or γ -sets of G, and two γ -sets are adjacent in G ( γ ) if they differ by a single vertex and the two different vertices are adjacent in G. In this paper, we consider γ -graphs of trees. We develop an algorithm for determining the γ -graph of a tree, characterize which trees are γ -graphs of trees, and further comment on the structure of γ -graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.

scholarly journals Spectra of some Operations on Graphs

2016 ◽  
Vol 11 (9) ◽  
pp. 5654-5660
Author(s):  
Essam EI Seidy ◽  
Salah ElDin Hussein ◽  
Atef Abo Elkher
Keyword(s):  
Direct Product ◽  
Cartesian Product ◽  
Simple Graph ◽  
Simple Graphs ◽  
Cartesian Product Of Graphs ◽  
Vertex Set ◽  
Join Of Graphs ◽  
Product Of Graphs

In this paper, we consider a finite undirected and connected simple graph G(E, V) with vertex set V(G) and edge set E(G).We introduced a new computes the spectra of some operations on simple graphs [union of disjoint graphs, join of graphs, cartesian product of graphs, strong cartesian product of graphs, direct product of graphs].

s-strongly perfect cartesian product of graphs

1992 ◽  
Vol 16 (4) ◽  
pp. 297-303
Author(s):  
Elefterie Olaru ◽  
Eugen M??ndrescu
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

scholarly journals On Topologies Induced by Graphs Under Some Unary and Binary Operations

2019 ◽  
Vol 12 (2) ◽  
pp. 499-505
Author(s):  
Caen Grace Sarona Nianga ◽  
Sergio R. Canoy Jr.
Keyword(s):  
Undirected Graph ◽  
Cartesian Product ◽  
Binary Operations ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺^2(𝑣) = {u ϵ V(G):  𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G^2[A] = V(G) \ N_G^2 [A] where N_G^2 [A] = A ∪ {v ϵ V(G):  𝑁_𝐺^2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs.

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

Sign in / Sign up

Export Citation Format

Share Document