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

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

ISBN-10: 1931233985

ISBN-13: 9781931233989

This booklet provides a few new sorts of Fuzzy and Neutrosophic types that could research difficulties in a progressive means. the hot notions of bigraphs, bimatrices and their generalizations are used to construct those types with a purpose to be worthy to research time based 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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15: A separable bigraph consists of two or more non separable bisubgraphs. Each of the largest non separable sub bigraphs is called a biblock or bicomponent. We just illustrate this by the following example. 29: Let G = G1 ∪ G2 be a separable bigraph given by the following figure. 29 58 G = G1 ∪ G2 where V (G1) = {v1, v2, v3, …, v7} and V(G2) = { v'1, v'2, v'3, …, v'9}. The separate graphs of G1 and G2 are given by the following figure. H and K are subbigraphs of G = G1 ∪ G2. 29d. 29d Now we can extend the notion of bigraphs to trigraphs, quadruple graphs and so on say to n-graphs n ≥ 2.

The disjoint bigraph G = G1 ∪ G2 is disconnected. A bigraph G = G1 ∪ G2 is connected if both G1 and G2 are connected and the bigraph is vertex glued bigraph or edge glued bigraph or a subgraph glued bigraph. A bitree is a connected bigraph without any circuits. Now we proceed on to define cut-set in a connected bigraph. 13: In a connected bigraph G = G1 ∪ G2 a bicut set is a set of edges (together with their end vertices) whose removal from G = G1 ∪ G2 leaves both the graphs G1 and G2 disconnected, provided removal of no proper subset of these edges disconnects G.

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


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