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

ISBN-10: 1931233985

ISBN-13: 9781931233989

This publication offers a few new kinds of Fuzzy and Neutrosophic types that could study difficulties in a innovative means. the hot notions of bigraphs, bimatrices and their generalizations are used to construct those versions so one can be necessary to research time established difficulties or difficulties which want 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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5: If G = G1 ∪ G2 be a disjoint bigraph such that G1 = G2C then G is a self complementary bigraph. Proof: Follows directly by the definition and the fact the bigraph is disjoint. We need to work more only when the bigraphs are not disjoint. 10: Let G = G1 ∪ G2 be a bigraph and v ∈ V (G1) ∪ V (G2). The number of edges incident at v in G = G1 ∪ G2 is called the degree (or valency) of the vertex v in G and is denoted by dG (v) or simply by d(v) when G requires no explicit reference. A graph G is called K regular if every vertex has degree K.

24 51 The concept of dibigraph will prove a major role in the applications of bimodels both in fuzzy theory and neutrosophic theory. Finally we give the matrix representations of bigraphs. First we give the simple bigraph and the related adjacency bimatrix. 25: Let G = G1 ∪ G2 be the bigraph G = G1 ∪ G2 in which both G1 and G2 are simple. 25. 25 The adjacency bimatrix of the bigraph is a mixed square bimatrix given as X = X1 ∪ X2 where v1 v1 ⎡0 v2 ⎢⎢1 X1 = v3 ⎢0 ⎢ v4 ⎢0 v5 ⎢1 ⎢ v6 ⎢⎣1 v2 v3 v4 v5 v6 1 0 0 1 1⎤ 0 0 1 1 0 ⎥⎥ 0 0 1 0 0⎥ ⎥ 1 1 0 1 1⎥ 1 0 1 0 1⎥ ⎥ 0 0 1 1 0 ⎥⎦ 52 v1' v1 ⎡0 v2 ⎢1 X2 = ⎢ v3 ⎢0 ⎢ v4 ⎢0 v5 ⎢⎣1 v '2 v 3' v4' v 5' 1 0 0 1⎤ 0 1 0 1 ⎥⎥ 1 0 1 1⎥ ⎥ 0 1 0 0⎥ 1 1 0 0 ⎥⎦ Thus X = X1 ∪ X2 is the adjacency bimatrix.

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.