Integrity of total transformation graphs

Abstract

A communication network can be considered to be highly vulnerable to disruption if the failure of few members (nodes or links) can result in no members being able to communicate with very many others. These communication networks can be modeled through graphs, and we have several graph-theoretic parameters (viz., connectivity, edge-connectivity, tenacity etc.,) to describe the stability of graphs. But, these parameters are not sufficient to study stability of graphs. This leads to the concept of integrity of a graph. The integrity of a graph will consider both the damage and the maximum possible capacity of communication corresponding to the maximum damage to the network. Therefore, we discuss the integrity of total transformation graphs which can help us to reconstruct the given network in such a way that it is more stable than the earlier one. If the network is modeled through total transformation graphs, then there will be increase in the number of nodes and links between the nodes in the obtained network which automatically cause the increase in the stability of the network. In this paper, we obtain the integrity of total transformation graphs of some special class of graphs. Further, we present bounds of integrity of some total transformation graphs of a graph in terms of number of vertices, number of edges and integrity of some derived graph appears as induced subgraph. The expression for integrity of total graph of cycle which was given by Qingfang Ye contained an error. We give correct version of it. In addition, we compute integrity of book graphs.