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

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

This publication offers a few new varieties of Fuzzy and Neutrosophic types which may examine difficulties in a progressive method. the hot notions of bigraphs, bimatrices and their generalizations are used to construct those versions with a purpose to be precious to investigate time established difficulties or difficulties which desire stage-by-stage comparability of greater than specialists. The types expressed the following might be regarded as generalizations of Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps.

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14a. 14b. 14b The study of conditions making a bigraph into a bipartite graph is an interesting one. We say as in case of graphs in bigraphs also the following. Since a bigraph can be realized as the ‘union’ of two graphs here the ‘union’ is distinctly different from terminology union of graphs used. The symbol just denotes only connection or union as subsets. Thus a bigraph G can be realized as G = (V (G1), E (G1), IG1 ) ∪ (V (G2), E (G2), IG 2 ) where Vi (G) is a nonempty set (i = 1, 2) and Ei (G) is a set disjoint from Vi (G) for i = 1, 2 and IG1 and IG 2 are incidence maps that associates with each element of Ei (G) an unordered pair of elements of Vi (G) i = 1, 2.

31c Infact G = G1 ∪ G2 ∪ G3 is a trigraph which is connected. Infact G is not a disjoint trigraph. 62 Now we proceed on to give an example of just a disjoint trigraph. 32. 32c Clearly this trigraph is a disjoint trigraph and the bigraph in it is a edge glued bigraph. Thus we have seen three types of trigraphs. In fact all properties of graphs can be proved with suitable and necessary modification for every trigraph is also a graph. Now we define a quadgraph G = G1 ∪ G2 ∪ G3 ∪ G4 as the union of four graphs G1, G2, G3 and G4 such that the vertex set of any one of them is not a proper subset of the other.

12c Here H = {v1, …, v6} ∪ {v'1, v'2, …, v'6} H1 = {v1, …, v6} and H2 = {v'1, v'2, …, v'6}. 37 Now we proceed on to give the three types of bimatrices associated with the bigraphs. We now show by the following example that a bipartite graph in general is not a bigraph. 13: Consider the bipartite graph B (G) of G given by the following figure. 13 Clearly this is not a Bigraph. 14. 14 = G1 ∪ G2 = {x1, x2, x3, y1, y2, y3} ∪ {x3 y3 x4 y4}. 14a. 14b. 14b The study of conditions making a bigraph into a bipartite graph is an interesting one.

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