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In this paper, we define d 2 -degree and total d 2 -degree of a vertex in fuzzy graphs. Further we study (2, k)-regularity and totally (2, k)-regularity of fuzzy graphs and the relation between (2, k)regularity and totally (2, k)-regularity. Also we study (2, k)-regularity on path on four vertices, barbell graph B n,n , n > 1 and cycle C n with some specific membership functions.doi:10.26708/ijmsc.2014.2.4.07 fatcat:ezarqrclsvaspiogwuhxp2qice