On $(r,2,k)$-Regular Fuzzy Graphs

Abstract

In this paper, $(r,2,k)$-regular fuzzy graphs and totally $(r,2,k)$-regular fuzzy graphs are defined and $(r,2,k)$-regular fuzzy graphs and totally $(r,2,k)$-regular fuzzy graphs are compared through various examples. A necessary and sufficient condition under which they are equivalent is provided. Also, $(r,2,k)$-regularity on some fuzzy graphs whose underlying crisp graphs are a path on four vertices, a Barbell graph $B_{n,n}$ $(n>1)$ and a cycle is studied with some specific membership functions.