Pinar Dundar

On the neighbour-integrity of sequential joined graphs

In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. We consider that two graphs have the same connectivity; but the order of their largest components are not equal. Then, these two graph must be different in respect to stability. How can we measure that property? Many graph theoretical parameters have been used in the past to describe the stability of communication networks. New parameters take into account what remains after the graph is disconnected. Several of these deal with the two fundamental questions. How many vertices can still communicate? How difficult is it to reconnect the graph? The neighbour-integrity of a graph is one measure of graph vulnerability. In the neighbour-integrity, it is considered that any failured vertex effects its neighbour vertices. In this work, we consider the neighbour-integrity number of sequential joined graphs that represent communications network. We give the theorems on this classes of graphs.

Stability measures of some static interconnection networks

It is important that a communication service has to service dependability by high level. Many affairs cause failures in a network. Destroying nodes or links in communication network, cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points, human error or accident and can interrupt service for long periods of time. At the beginning a communication network, requiring greater degree of stability or less vulnerability. In this work, various stability measures of a communication network are defined and the stability measures of some static interconnection networks which are known long times and w-star networks that are a new graph class, are given.It is important that a communication service has to service dependability by high level. Many affairs cause failures in a network. Destroying nodes or links in communication network, cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points, human error or accident and can interrupt service for long periods of time. At the beginning a communication network, requiring greater degree of stability or less vulnerability. In this work, various stability measures of a communication network are defined and the stability measures of some static interconnection networks which are known long times and w-star networks that are a new graph class, are given.

Integrity of Total Graphs via Certain Parameters

Communication networks have been characterized by high levels of service reliability. Links cuts, node interruptions, software errors or hardware failures, and transmission failures at various points can interrupt service for long periods of time. In communication networks, greater degrees of stability or less vulnerability is required. The vulnerability of communication network measures the resistance of the network to the disruption of operation after the failure of certain stations or communication links. If we think of a graph G as modeling a network, many graph-theoretic parameters can be used to describe the stability of communication networks, including connectivity, integrity, and tenacity. We consider two graphs with the same connectivity, but with unequal orders of theirs largest components. Then these two graphs must be different in respect to stability. How can we measure that property? The idea behind the answer is the concept of integrity, which is different from connectivity. Total graphs constitute a large class of graphs. In this paper, we study the integrity of total graphs via some graph parameters.

The neighbour-integrity of boolean graphs and its compounds

Communication services have traditionally been characterized by high levels of service dependability. Today cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points can cause interrupted service for long periods of time. Large amounts of data transmitted through fiber optic and satellite communication networks, requires greater degrees of stability or less vulnerability. The neighbour integrity is a vulnerability measure of a graph which is considered as a communications network model. This paper presents neighbour integrity of a Boolean graph. Also, theorems are given for some compounds of Boolean graphs which are obtained by using the graph operations.

Finding a fault tolerant routing on neighbour-faulty hypercube

It is important for a communication network to route data efficiently among nodes. Efficient routing can be achieved by using node-disjoint paths. Routing by node-disjoint paths among nodes cannot only avoid communication bottlenecks, and thus increase the efficiency of message transmission, and also provide alternative paths in case of node failures. The one-to-one routing constructs the maximum number of node-disjoint paths in the network between the two given nodes. Optimal algorithms have proved sufficiency in designing efficient and fault tolerant routing algorithms on hypercube networks. In this article, firstly, we give the concepts of neighbourhood and accessibility in a graph, which is a model of a communication network. After, we give a fault tolerant routing algorithm using cube algebra for faulty hypercube that is proceeded from neighbour notion. E-mail: elginkilic@sci.ege.edu.tr

Total accessibility number of graphs

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.

The Center Colorıng of a Graph

Abstract For a nontrivial connected graph G, center coloring is a kind of coloring that is to color the vertices of a graph G is such a way that if vertices have different distance from the center then they must receive different colors. Two adjacent vertices can receive the same color. The number of colors required of such a coloring is called center coloring number Cc (G) of G. [7] This coloring can be applied to hierarchy problems to find the number of structures, people, criteria and comparisons, etc. Moreover it can be applied to earthquake motion problems to find the number of settlements that are affected by an earthquake. The center coloring number of some well-known classes of graphs are determined and several bounds are established for the center coloring number of a graph in terms of other graphical parameters.

The Average Covering Number of a Graph

There are occasions when an average value of a graph parameter gives more useful information than the basic global value. In this paper, we introduce the concept of the average covering number of a graph (the covering number of a graph is the minimum number of vertices in a set with the property that every edge has a vertex in the set). We establish relationships between the average covering number and some other graph parameters, find the extreme values of the average covering number among all graphs of a given order, and find the average covering number for some families of graphs.

The Average-Dominating of a graph

The vulnerability is one of the most important concepts in network design. Vulnerability can be considered as the resistance of the network after any breakdown in its nodes or links. Since any network can be modelled by a graph, vulnerability parameters on graphs also work on network types. The vertices and the edges of a graph correspond to the processors and the links of the network, respectively. The measure parameters about the vulnerability of a connected graph which are mostly used and known are based on the Neighbourhood concept. Neighbour-integrity, edge-integrity and accessibility number are some of these parameters. In this work we defined and examined the Average-Dominating of a connected graph as a new global connectivity measure. It takes account the neighbourhoods of all pairs of vertices. First we define the pair dominating of any two vertices such as u and v in a graph as to be the maximum number of vertices which dominate both u and v in G. It is denoted by D(u,v). Then latter we define the average dominating of G denoted by D(G) as, where G is a graph of order n.