On average vulnerability parameters in graph theory

In a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. This study introduces a new graph parameter. For an edge e of a graph G, the lower edge covering number, denoted by , is the minimum cardinality of a edge covering set of G that contains e. The average edge covering number of a graph G is, .Where , will denote the sum over all edges of G. In this thesis, we give the relationships between average edge covering number and some graph parameters and find the average covering number for some families of graphs. Also an algorithm for computing for the average edge covering number of any graph is given. Key Words: vulnerability, network design and communication, average lower domination number, vertex cover, edge cover, average vertex cover