Regular fuzzy graphs and totally regular fuzzy graphs

The fuzzy graph theory was introduced by Azriel Rosenfeld in 1975. Fuzzy graph theory is useful in mobile communication. In the transmission of signals this will help to find the minimum length of the path between any two mobile towers which can transmit maximum signal strength. Let the mobile towers denote the vertices of the fuzzy graph having the maximum signal strength as its membership value and the membership value of the edges are determined by the strength of the signals between these towers. In this paper, we consider two of the special fuzzy graph, regular fuzzy graph and totally regular fuzzy graph, that their underlying graph is cycle. We obtain conditions under which a fuzzy graph with cycle underlying graph is regular fuzzy graph and totally regular fuzzy graph. Also, we get the number of bridges and cut vertices in these fuzzy.