By W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral

This publication offers a few new varieties of Fuzzy and Neutrosophic versions which could study difficulties in a progressive method. the recent notions of bigraphs, bimatrices and their generalizations are used to construct those types to be able to be worthy to research time established difficulties or difficulties which desire stage-by-stage comparability of greater than specialists. The types expressed the following could be regarded as generalizations of Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps.

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23. 23a. 23a Now we proceed on to define the notion of directed bigraph. 11: A directed bigraph G = G1 ∪ G2 is a pair of ordered triple {(V (G1), A (G1), I G1 ) , (V (G2), A (G2), I G2 )} where V (G1) and V (G2) are non empty proper sets of V (G) called the set of vertices of G = G1 ∪ G2. A(G1) and A (G2) is a set disjoint from V (G1) and V(G2) respectively called the set of arcs of G1 and G2 and I G1 and I G2 are incidence map that associates with each arc of G1 and G2 an ordered pair of vertices of G1 and G2 respectively.

It is to be noted that E1 (G1) ⊂/ E2 (G2) or V1 (G1) ⊆/ V2 (G2), IG1 ⊆/ IG 2 and E2 (G2) ⊂/ E1 (G1) or V2 (G2) ⊂/ V1 (G1), IG1 ⊆/ IG 2 . e. in general E1 (G1 ) ∩ E 2 (G 2 ) ≠ φ ⎫ ⎪ (I) I1 (G1 ) ∩ I 2 (G 2 ) ≠ φ and ⎬ ⎪ V1 (G1 ) ∩ V2 (G 2 ) ≠ φ ⎭ But nothing prevents us from having 39 V1 (G1 ) ∩ V2 (G 2 ) = φ or ⎫ ⎪ E1 (G1 ) ∩ E 2 (G 2 ) = φ and ⎬ ⎪ IG1 ∩ IG 2 = φ. e. they are union of disjoint graphs. Thus union of any two graphs is a bigraph provided the union does not take place between the graph and its subgraph.

10b Hence the claim. We can have bigraphs glued by more than one edge or by a subgraph which happen to be the subgraph of both the graphs. We define it as follows. 5: Let G = G1 ∪ G2 be a bigraph. Suppose the bigraphs is glued such that it has a subgraph (with more than one vertex and more than one edge) in common then we define this bigraph as a bigraph glued by a strong subgraph or a strong subgraph glued bigraph. 33 We illustrate this by the following example. 11: G = G1 ∪ G2 be a bigraph given by the following figure.

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Applications of Bimatrices to Some Fuzzy and Neutrosophic by W. B. Vasantha Kandasamy, Florentin Smarandache, K.
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