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

ISBN-10: 1931233985

ISBN-13: 9781931233989

This e-book provides a few new sorts of Fuzzy and Neutrosophic types that can learn difficulties in a progressive approach. the hot notions of bigraphs, bimatrices and their generalizations are used to construct those versions that allows you to be precious to research time based difficulties or difficulties which want stage-by-stage comparability of greater than specialists. The versions expressed right here might 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**

**Sample text**

Study in that direction may throw light on the properties of the bigraph. 8: Let G = G1 ∪ G2 be a bigraph we say the bigraph is simple if G is simple, pseudo simple if both G1 and G2 is simple but G is not simple. The simple bigraph G is said to be bicomplete if every pair of distinct vertices of G are adjacent in G. 4: If G1 and G2 are complete the bigraph G = G1 ∪ G2 need not in general be complete. Proof: If G is a disjoint bigraph even if G1 and G2 are complete G need not be complete. This clear from the following example.

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.

It is to be noted that given any weighted bimatrix which is symmetric one can get back to the bigraph and given the bigraph we always have a bimatrix which weighed can be associated with it. Suppose G = G1 ∪ G2 is a dibigraph then we can have the incident bimatrix associated with it. The incidence matrix of a digraph with n vertices and e edges and no self loops. We show by the following example the incidence bimatrix associated with the dibigraph. 27. 27 The incidence bimatrix of the bigraph. v1 v2 v3 v4 v5 v6 a1 ⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢1 ⎢0 ⎢ ⎣⎢1 b1 0 0 0 1 0 1 c1 0 0 0 1 1 0 d1 1 0 0 0 1 0 e1 0 1 0 1 0 0 f1 1 1 0 0 0 0 g1 h1 a2 b2 0 0⎤ ' v ⎡ 1 1 0 1 1 ⎥⎥ v 2' ⎢⎢1 1 0 1⎥ ∪ v 3' ⎢0 1 ⎥ 0 0⎥ ⎢ v 4' ⎢0 0 ⎥ 1 0 v 5' ⎢⎣0 0 ⎥ 0 0 ⎦⎥ c2 0 0 1 1 0 d2 1 0 0 1 1 e2 0⎤ 0 ⎥⎥ 0⎥ ⎥ 1⎥ 0 ⎥⎦ The incidence bimatrix associated with the bigraph is a mixed rectangular bimatrix.

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

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