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.
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