The main purpose of this paper is to introduce the notion of bipolar f-morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We study the action of bipolar f-morphism on bipolar fuzzy graphs and derive some elegant results on weak and co-weak isomorphism. Also, we define d(2)-degree and total d(2)-degree of a vertex in bipolar fuzzy graphs and study (2, k)-regularity and totally (2, k)-regularity of bipolar fuzzy graphs. d(2)-degree and total d(2)-degree of a vertex in bipolar fuzzy graphs are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition these properties will also be helpful to study large bipolar fuzzy graph as a combination of small, bipolar fuzzy graphs and to derive its properties from those of the smaller ones.
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