Cyclic Connectivity Index of Fuzzy Graphs

A parameter is a numerical or other measurable factor whose values characterize a system. Connectivity parameters have indispensable role in the analysis of connectivity of networks. Strength of a cycle in an unweighted graph is always one. But, in a fuzzy graph, strengths of cycles may vary even for a given pair of vertices. Cyclic reachability is a property that determines the overall connectedness of a network. This article introduces two connectivity parameters namely, cyclic connectivity index (CCI) and average CCI (ACCI) of fuzzy graphs, which can be used to represent the cyclic reachability. CCI of fuzzy graph theoretic structures, such as trees, blocks, theta-fuzzy graphs, and complete fuzzy graphs, are discussed. Vertices of a fuzzy graph are classified into three categories in terms of ACCI and their characterizations are obtained. Three algorithms are proposed. One of them is to help find CCI and ACCI of a given fuzzy graph. Another helps to identify the nature of vertices and the third to enhance the existing ACCI of a fuzzy graph. Also, future directions in the study of CCI and ACCI are proposed.