When the nodes or links of communication networks are destroyed,
its effectiveness decreases. Thus, we must design the communication network as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network. A graph is considered as a modeling network, many graph theoretic parameters have been ušed to describe the stability of communication networks, including connectivity, integrity, tenacity. Several of these deal with two fundamental questions about the resulting graph. How many vertices can still communicate? How difficult is it to reconnect the graph? Stability numbers of a graph measure its durability respect to break down. The neighbour-integrity of a graph is a measure of graph vulnerability. In the neighbour-integrity, it is considered that any failure vertex effects its neighbour vertices. In this work, we define the accessible sets and accessibility number and we consider the neighbour-integrity of Generalised Petersen graphs and the relation with its accessibility number.
The centre of a communications network is a vertex set. The distances between every vertex in the centre set and all other vertices of the network are minimal. In some cases, the centre of the network can be a path, which includes a desired number of vertices. This centre is called a path centre of the network. In this paper, we aim to find a path centre of a given network with the needed number of vertices. We give the distance measures of the network and represent an algorithm searching the path centre of the network.
Článok sa zaoberá problematikou použitia generátorov pseudonáhodných čísiel v numerických metódach štatistickej fyziky. Na začiatku stručne oboznamuje s históriou metód Monte Carlo. Ďalej obsahuje popis jednoduchého algoritmu, ktorý pomocou náhodných čísiel vypočítá Ludolfovo číslo. V poslednej časti je predložený prehĺad známych generátorov a posúdenie ich vhodnosti pro použitie v metódach Monte Carlo., Michal Kaňok., and Obsahuje seznam literatury
Postačí čtyři barvy na obarvení každé rovinné mapy obsahující jistý počet států? Tato zdánlivě nevinná otázka napadla Francise Guthrieho při barvení mapy anglických hrabství. Jeho bratr Frederick položil dne 23. 10. 1852 stejnou otázku svému profesoru Augustu de Morganovi. Tak vznikl slavný "problém čtyř barev“. Po nezbytných upřesněních pojmu rovinná mapa atd. se zdařilo daný problém "přeložit“ do rozvíjející se matematické disciplíny - teorie grafů. Následovala dlouhá historie řešení, která trvala přes sto let a byla provázena i mnoha omyly (více viz [1,3]). Nakonec, již v sedmdesátých letech minulého století, bylo zapotřebí prověřit dlouhý seznam "map“, tzv. nevyhnutelnou množinu ireducibilních konfigurací, obsahující 1 936 prvků. K tomu přistoupili Apell, Haken a Koch - využili hned tři počítače firmy IBM. Příprava metod a programu jim zabrala tři a půl roku a další půlrok si vyžádala práce s počítači. Dílo bylo dokončeno 21. června 1976. Závěr zněl: čtyři barvy stačí!, Jaroslav Hora., and Obsahuje seznam literatury
The vulnerability of the communication network measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. Cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points can cause interrupt service for long periods of time. High levels of service dependability have traditionally characterised communication services. In communication networks, requiring greater degrees of stability or less vulnerability. If we think of graph G as modelling a network, the neighbour-integrity and edge-neighbour-integrity of a graph, which are considered as the neighbour vulnerability, are two measures of graph vulnerability. In the neighbour-integrity, it is considered that any failure vertex affects its neighbour vertices. In the edge-neighbour-integrity it is consider that any failure edge affects its neighbour edges.
In this paper we study classes of recursive graphs that are used to design communication networks and represent the molecular structure, and we show neighbour-integrity (vertex and edge) among the recursive graphs.
A network begnis losing nodes or liriks, or there inay be a loss in its
effectiveness. Thus, the communication network must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network. Stability numbers of a communication network measure its durability with respect to a break down. If we consider a graph as modelling of a communication network, connectivity is an important measure of reliability or stability of a graph, but not enough. Integrity is a new measurement of stability. It takes into consideration the number of vertices of the remaining cornponents after some disruption. Also the edge-integrity is defined. In this paper, we study integrity (or vertex-integrity) and edge-integrity of Double Star Graphs and some of its cornpounds.
The vulnerability value of a communication network shows the resistance of the network after the disruption of some centres or connection lines until the communication breakdown. In a network, as the number of centres belonging to sub networks changes, the vulnerability of the network also changes and requires greater degrees of stability or less vulnerability. If the communication network is modelled by a graph G, deterministic measures tend to provide a worst-case analysis of some aspects of the overall disconnection process. Differently from other measures, in the neighbour-integrity is considered that any failure vertex affects its neighbour vertices. Neighbour-integrity is very important measure in stability of security networks and spy networks. It replies three questions: How many vertices can still communicate? How difficult is it to reconnect the graph? How can we design an optimal network?
In this paper we discuss the concept of neighbour-integrity. Firstly, we give some definitions and notation and then we calculate some stability numbers of two-dimensional mesh and torus graphs, which are ušed in computer sciences.
When a network begins losing nodes or links there is, eventually, a loss in its effectiveness. Thus, a communication network must be constructed to be as stable as possible, not only with respect to tlie initial disruption, but also with respect to the possible reconstruction of the network. When any disruption happens in a cornmunication network two questions are considered: How many vertices can still communicate? How difficult is it to reconnect the network? If a graph is considered as a modeling network, then the above questions can be answered by the graphs. Many graph parameters have been used to deseribe the stability of communication networks, including connectivity, integrity, and tougliness and the binding number. The thorny graphs are special classes of graphs that represent some static interconnection networks. In tliis work, we have given the tenacity of thorny graphs of static interconnection networks.
One of the most important problems in communication network design is the stability of network after any disruption of stations or links. Since a network can be modeled by a graph, this concept is examined under the view of vulnerability of graphs. There are many vulnerability measures that were defined in this sense. In recent years, measures have been defined over some vertices or edges having specific properties. These measures can be considered to be a second type of measures. Here we define a new measure of the second type called the total accessibility. This measure is based on accessible sets of a graph. In our study we give the total accessibility number of well known graph models such as Pn, Cn, Km,n, W1,n, K1,n. We also examine this new measure under operations on graphs. A simple algorithm, which calculates the total accessibility number of graphs, is given. We observe that when any two graphs of the same size are compared in stability, it is inferred that the graph of higher total accessibility number is more stable than the other one. All the graphs considered in this paper are undirected, loopless and connected.
In this paper we extend the notion of weak degree domination in graphs to hypergraphs and find relationships among the domination number, the weak edge-degree domination number, the independent domination number and the independence number of a given hypergraph.