On Vulnerability of Power and Total Graphs

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 the network begins losing communication links or processors, then there is a loss in its effectiveness. Thus, communication networks must be so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters like toughness, integrity, scattering number, tenacity and rupture degree. Power graphs and total graphs constitute a large class of graphs and which are widely used in systems ranging from large supercomputers to small embedded systems-on-a-chip. In this paper, we firstly give the exact values for the integrity and toughness of powers of paths. After that, the vulnerability parameters such as integrity, toughness, rupture degree of total graphs of some special graphs are calculated. Finally, the relationships between some vulnerability parameters, namely the integrity, toughness, scattering number, tenacity and rupture degree are established. Key–Words: Vulnerability, Integrity, Toughness, Scattering number, Tenacity, Rupture degree, Total graph, Power graph.