Building Graphs with Maximum Connectivity

Abstract

The increasing data volume and the algorithms complexity in recent years has made topics such as cloud computing extremely important. Meanwhile, improving the communication parameters in cloud computing has led to improvements such as the increase in efficiency, and the reduction of execution time and the energy consumption. The concept of communication can be treated as mathematical modeling with graphs, where the network requirements are evaluated with the graph parameters. Parameters such as connectivity, toughness, tenacity, and graph diameter are commonly used in cloud computing, social networks, networks security, etc. For example, connectivity is one of the major graph parameters that defines the extent of the graph's vulnerability. This parameter is evaluated in two vertex and edge modes. Creating graphs with a maximum vertex connectivity value was proposed by Harary and is referred to as Harary graphs. In this paper, we explore other graphs that have a different structure from Harary graphs and we make an attempt to improve other parameters such as graph diameter and toughness in the proposed graphs. In addition to cloud computing, the theoretical results of the proposed models are applicable in various fields such as social networks, network security, electronic circuit design, geography and urban design, bioinformatics and more.