By John N. Mordeson, Premchand S. Nair

The booklet bargains with fuzzy graph conception, fuzzy topology, fuzzy geometry, and fuzzy summary algebra. It offers the innovations of fuzzy arithmetic with purposes to engineering, computing device technology, and arithmetic. during this moment variation the bankruptcy on geometry is accelerated and comprises effects at the measure of adjacency of 2 areas and the measure of surroundedness of a area via one other. purposes to electronic polygons and snapshot description also are given. additionally, the recent version comprises effects on photo enhancement and thresholding through optimization of fuzzy compactness in addition to effects bearing on a Hausdorff distance among fuzzy subsets. the most up-tp-date paintings at the answer of nonlinear platforms of fuzzy intersection equations of fuzzy singletons has been additional to the bankruptcy on algebra. The e-book is written with engineers and desktop scientists in brain, however it may also function a examine advisor to mathematicians because it consists of present effects.

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**Example text**

V (n RI(c,e)) = (1/4AOn c,dEC cEC 0)V(OAOAO) = 0, V} ( A R2(c,e)) = (1/4n1/8n1/8)V(1/4AOAO) = 1/8, eOC, cEC and V ( A Rk(c, e)) = (1/4 n 1/8 n 1/8) V (1/4 n 1/8 n 1/8) = 1/8 for cEC k > 3. Hence C is a fuzzy cluster of order 1, but not of order k for k > 2. Bridges and Cut Vertices Let G = (A, R) be a fuzzy graph, let x, y be any two distinct vertices, and let G' be the partial fuzzy subgraph of C obtained by deleting the 26 2. FUZZY GRAPHS edge (x, y). That is, G' _ (A, R'), where R'(x. y) = 0 and k = R for all other pairs.

26. Hence (A1 x A2, E1 E2) is the Cartesian product of partial fuzzy subgraphs (B2, F2) of Gi, i = 1, 2. In fact, these Bi and F2 (i = 1, 2) are constant membership functions with membership value 1/2. Union and Join Consider the union G = G1 U G2 of two graphs G1 = (V1, X,) and G2 = (V2, X2), [16]. Then G = (V1 U V2, X1 U X2). Let A; be a fuzzy subset of V and Ei a fuzzy subset of Xi, i = 1, 2. Define the fuzzy subsets A, U A2 of V, U V2 and E1 U E2 of X 1 UX2 as follows: (A,UA2)(u)=A,(u)ifnE V,\V2i(A,UA2)(u)=A2(u)if uE V2\V1i and (Al U A2) (U) = A, (u) V A2(u) if u E VI n V2; (E1 U E2)(uv) = Ei(uv) if uv E X1 \ X2, (E1 U E2)(uv) = E2(uv) if uv E X2 \ X1, and (E1 U E2) (uv) = E,(uv) V E2(uv) if uv E X1 n X2.

Y,,,} U {wihl(i,h) E I} is any solution to (i), (ii), and (iii), then {xI .... , x } U {zjkl (j. k) E J} U {yl, ... , ym } U {wiht(i, h) E I} is also a solution and, in fact, zjk is the smallest possible z,7k and wih is the smallest possible wih. Fix such a solution and define the fuzzy subsets A1i A2, El . , m; E2(v2jv2k) = zjk for j, k such that v2jv2k E X2; Ei(vlivlh) = wih for i,h such that vlivlh E X1. , n. ,n} < A2(v2j) A A2(v2k). Hence E2(v2jv2k) < A2(v23)AA2(v2k). Thus (A2, E2) is a partial fuzzy subgraph of G2.