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

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

This booklet offers a few new kinds of Fuzzy and Neutrosophic types which may study difficulties in a innovative means. the hot notions of bigraphs, bimatrices and their generalizations are used to construct those versions for you to be necessary to investigate time established difficulties or difficulties which desire stage-by-stage comparability of greater than specialists. The types expressed the following will 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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22a. 22a Clearly G and G1 are distinct. 22b. 22b 49 Thus G = G1 ∪ G2 ≠ G1. 22c. 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.

11 Here G = G1 ∪ G2 = {v1, v 2, v 3, …, v8} ∪ { v'1, v'2, v'3, …, v'8}. The graphs associated with G1 and G2 are given by the following figure. 11b This bigraph has a subgraph in common given by the following figure. 11c Thus we have the following interesting result. 3: Let G = G1 ∪ G2 be a bigraph which is strong subgraph glued bigraph, then G is a vertex glued graph and a edge glued graph. Proof: Since single point or an edge is subgraph we see vertex glued bigraph and a edge glued bigraph are also subgraph glued bigraphs, but cannot be called as strong subgraph glued bigraph.

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.

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