Random walk on fuzzy graphs

Abstract

In this research, operations on fuzzy set theory is extended to the fuzzy adjacent matrix of the fuzzy graph. In this process many properties of fuzzy adjacent matrix is discussed. Fuzzy adjacent matrices is probably the most frequently used matrix representation of a fuzzy graph. Fuzzy graph is represented by fuzzy adjacent matrix. It can be manipulated in many ways than a non square matrix. Basic characteristics of the fuzzy adjacent matrix are highlighted. Under max-min and min-max composition several properties of fuzzy adjacent and diagonal degree matrices of fuzzy graph are found. The relation between fuzzy adjacent matrix and diagonal degree matrix of fuzzy graph is discussed. The fuzzy adjacent matrix provides the sound interpretation on the bounds. The energy of the fuzzy graph is calculated using the eigen values of fuzzy adjacent matrix. This enables to determine the bounds and spectra of the fuzzy graph. The condition for the existence of fuzzy walk in a fuzzy graph is captured. The edge sequence, composite fuzzy walk, inverse fuzzy walk and reachability of fuzzy graph is introduced. The necessary condition for existence of fuzzy path in fuzzy walk is given through fuzzy adjacent matrix. Variousresults characterizing fuzzy walk and its varieties with repect to fuzzy graph is given. The strength of a fuzzy walk through fuzzy adjacent matrix is established. The number of edge sequence with reference to fuzzy adjacent matrix under max min composition is proposed. The equivalence relation on fuzzy path is interpreted. The existence of random walk on fuzzy graph is determined with respect to the transition matrix. The random walk is classified into states. Regardless of the original state of a MC, the chain will not enter into an absorbing state in a finite number of steps. The long term trend that depends on the initial state is evaluated. The n step random walk on fuzzy graph is given through max-min and min max composition. The stationary distribution of random walk is found. newline