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 e-book supplies a few new different types of Fuzzy and Neutrosophic versions that could study difficulties in a progressive manner. the recent notions of bigraphs, bimatrices and their generalizations are used to construct those types so as to be important to investigate time established difficulties or difficulties which want stage-by-stage comparability of greater than specialists. The versions expressed the following should be regarded as generalizations of Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps.

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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. We say as in case of graphs in bigraphs also the following.

However the graph G2 has no pendent vertex or an isolated vertex. Now we give an example of a bigraph G = G1 ∪ G2 where G1 has both isolated vertex and pendent vertex but G2, the graph has no pendent vertex or isolated vertex. This sort of bigraph G = G1 ∪ G2 is said to have pseudo pendent vertex and pseudo isolated vertex. The following bigraph is an example of a bigraph with pseudo isolated vertex and pseudo pendent vertex. 21. 21 with G1 = {u1, …, u7} and G2 = {u'1, u'2, u'3, u'4}. 21b. 21b Clearly G1 has both an isolated vertex u1 and a pendent vertex u6 but the bigraph G = G1 ∪ G2 has no pendent vertex or isolated vertex.

22c Clearly G = G1 ∪ G2 ≠ G1. Thus the join of two graphs G1 ∨ G2 is not the same as the bigraph given by G = G1 ∪ G2. Also we can show that in general the direct product two graphs G1 and G2 cannot be got as a bigraph G1 ∪ G2. e. G1 × G2 ≠ G1 ∪ G2. For this is clear from the following example. 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.

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