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

ISBN-10: 1931233985

ISBN-13: 9781931233989

This booklet offers a few new varieties of Fuzzy and Neutrosophic types that can study difficulties in a innovative method. the recent notions of bigraphs, bimatrices and their generalizations are used to construct those versions in an effort to be beneficial to research time established difficulties or difficulties which want 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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Extra resources for Applications of Bimatrices to Some Fuzzy and Neutrosophic Models

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9 The two separate graphs are as follows. 9b This bigraph has three vertices in common. e. v1 = v'1, v3 = v'4 and v5 = v'3 but this bigraph has no edge in common. Hence the claim of the theorem. Now it is still interesting to note the following result. 2: A bigraph glued by even more than two vertices need not in general be glued by an edge. Proof: This result is proved by the following example. 10: Consider the bigraph G = G1 ∪ G2 given by the following figure. 10 That is 32 G1 = {v1, v 2, v 3, v 4} and G2 = {v'1, v'2, v'3, v'4, v'5, v'6, v'7,v'8}.

It is very interesting to note all bigraphs which are disjoint bigraphs are trivially separable bigraphs. Further all bigraphs glued by a vertex or single vertex glued bigraphs are separable. 6: Let G = G1 ∪ G2 which is a single vertex glued bigraph G is separable. Proof: Given G = G1 ∪ G2 is a bigraph which is a single vertex glued bigraph say let them be glued by the vertex vj = v'1 by removing that vertex, the bigraph becomes the separable bigraph. 12: A bigraph G = G1 ∪ G2 is connected if there is at least one path between every pair of vertices in G other wise G is disconnected.

G = G1 ∪ G2. 19 G = G1 ∪ G2 is a vertex glued bigraph. The vertex u2 and u'1 are glued. Clearly G1 is 3-regular and G2 is 4 regular. G = 46 G1 ∪ G2 is such that it has only one vertex u'1 = u2 in common and degree of u'1 = u2 is 5. So G is a 5-biregular bigraph. The notion of isolated vertex of the bigraph G = G1 ∪ G2 and pendent vertex of G can be defined as in case of graphs. The following example will illustrate the isolated vertex and pendent vertex of the bigraph G = G1 ∪ G2. 20. 20 G1 = {u1, u2, …, u7}.

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

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