Kiran R. Bhutani

Strong arcs in fuzzy graphs

An arc of a fuzzy graph is called strong if its weight is at least as great as the strength of connectedness of its end nodes when it is deleted. An arc is strong iff its weight is equal to the strength of connectedness of its end nodes. A bridge is strong, but a strong arc need not be a bridge. An arc of maximum weight is strong, but a strong arc need not have maximum weight. In a connected graph, there is a strong path (a path consisting of strong arcs) between any two nodes. A fuzzy graph is a fuzzy tree iff there is a unique strong path between any two of its nodes. In a fuzzy tree, an arc is strong iff it is a bridge, and a strong path between two nodes is a path of maximum strength between them.

Fuzzy Group Theory

Fuzzy Subsets and Fuzzy Subgroups.- Fuzzy Caleys Theorem and Fuzzy Lagranges Theorem.- Nilpotent, Commutator, and Solvable Fuzzy Subgroups.- Characterization of Certain Groups and Fuzzy Subgroups.- Free Fuzzy Subgroups and Fuzzy Subgroup Presentations.- Fuzzy Subgroups of Abelian Groups.- Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups.- Equivalence of Fuzzy Subgroups of Finite Abelian Groups.- Lattices of Fuzzy Subgroups.- Membership Functions From Similarity Relations.

On automorphisms of fuzzy graphs

Abstract We introduce some definitions for fuzzy graphs and provide examples to explain various notions introduced. We show that every fuzzy group can be imbedded in a fuzzy group of the group of automorphisms of some fuzzy graph.

On M-strong fuzzy graphs

The cartesian product and disjoint sum of graphs play a prominent role and have numerous interesting algebraic properties. In this note, we consider operations on fuzzy graphs under which M-strong property is preserved. If G1 and G2 are M-strong fuzzy graphs then we prove that G1 × G2, G1 [G2] and G1 + G2 are also M-strong but G1 ∪ G2 need not be M-strong. If G1 × G2 is M-strong then we show that at least one factor must be M-strong. We show that the product of a M-strong fuzzy graph G1 with a non-M-strong fuzzy graph G2 remains M-strong if and only if G2 satisfies special condition. For any fuzzy graph G, Gcc is the smallest M-strong fuzzy graph that contains G and G = Gcc if and only if G is M-strong. We further show that M-strong fuzzy graph G is a fuzzy tree if and only if the support(G) is a tree.

Fuzzy end nodes in fuzzy graphs

We define a fuzzy end node in a fuzzy graph, and show that no node can be both a cut node and a fuzzy end node. In a fuzzy tree, every node is either a cut node or a fuzzy end node, but the converse is not true. We also show that any nontrivial fuzzy tree has at least two fuzzy end nodes, and we characterize fuzzy cycles that have no cut nodes or no fuzzy end nodes.

Geodesies in Fuzzy Graphs

Abstract An arc (u, v) in a fuzzy graph H is called strong if its weight is at least as great as the strength of connectedness of u and v when (u, v) is deleted. A strong path is a path whose arcs are all strong, and a geodesic is a shortest strong path between its end nodes. Let S be a set of nodes. The closure (S) of S is the set of nodes that lie on geodesies between nodes of S. We say that S covers H if ( S ) = H . A minimal cover of H is called a basis. A (fuzzy) end node is a node that has only one strong neighbor. We prove that any cover of H contains all the end nodes of H , and that if H is a fuzzy tree, its set of end nodes is its unique basis, but not conversely. If H is connected, there is a strong path, and hence a geodesic, between any two nodes u, v of H. The length of a geodesic between u and v is called the g-distance dg(u, v). Using this concept of distance, we prove that the center of a fuzzy tree consists of either a single node or two nodes joined by a strong arc. A node is called a median of (u, v, w) if it lies on geodesies between u and v, v and w , and w and u. We prove that in a fuzzy tree, every triple of nodes has a unique median, but not conversely.

An application of fuzzy relations to image enhancement

Abstract When images are converted from one form to another they are subject to nonlinear transformations that reduce the output quality. This paper presents an image enhancement technique based on fuzzy relations that can be applied to improve the appearance of these images. A grey-scale image I with M rows and N columns is viewed as a weighted relation from a set X with M elements into a set Y with N elements with g ( x , y ) representing the grey-value at location ( x , y ) in I . An appropriate choice of two thresholds corresponding to pixels close to black and white pixels is made which partitions the image into crisp and fuzzy pixels. The fuzzy pixels correspond to pixels which are either black or white with a certain degree of membership between 0 and 1. The original image is transformed into an enhanced image by forming its dual complement. This is achieved by first constructing fuzzy sets σ x and σ y on X and Y respectively and applying these fuzzy sets to manipulate only the fuzzy pixels in I . After the entire transformation, the maximally fuzzy pixels, that is, pixels with membership value equal to 1 2 , get enhanced towards either black or white, depending on the neighborhood pixels of the segment to which they belong. The performance of the proposed technique is demonstrated on several images. Enhanced images obtained using other techniques are also presented for comparison.

THE RATIONALITY OF FUZZY CHOICE FUNCTIONS

If we assume that the preferences of a set of political actors are not cyclic, we would like to know if their collective choices are rationalizable. Given a fuzzy choice rule, do they collectively choose an alternative from the set of undominated alternatives? We consider necessary and sufficient conditions for a partially acyclic fuzzy choice function to be rationalizable. We find that certain fuzzy choice functions that satisfy conditions α and β are rationalizable. Furthermore, any fuzzy choice function that satisfies these two conditions also satisfies Arrow and Warp.

Two Approaches for Aggregation of Peer Group Topology in Hierarchical PNNI Networks

ABSTRACTWe propose two methods for aggregation of peer group topology in hierarchical ATM networks. Both proposed aggregation methods transform a given peer group into a star graph representation. Our first approach optimally preserves, in a least square sense, the original costs of routing through the peer group. Our second approach assigns a weighted vector to the nucleus of the Logical Group Node, which quantifies the error in the compact representation. The two schemes are dual, in the sense that the first is best suited for peergroups where traffic patterns are unpredictable, and the second is suited for peergroups where traffic patterns can be characterized. Both the proposed schemes are practical: For peer groups with nodes V, links E, and n border nodes B ⊂ V, the approaches run in O(n|V|log|V| + n|E|+ poly(n)) time. The size of the final representation is small (linear in the number of border nodes) and can be computed efficiently. The scalability of the proposed algorithms makes them well-suited...

Dissimilarity measures between fuzzy sets or fuzzy structures

Two simple dissimilarity measures between fuzzy subsets of a finite set S are defined, and it is shown that both of the measures are metrics. The measures can also be generalized to fuzzy structures, such as fuzzy graphs, defined on S ; the graph generalization is closely related to a recently defined measure of graph dissimilarity.