Premchand S. Nair

Fuzzy Graphs and Fuzzy Hypergraphs

Fuzzy Subsets: Fuzzy Relations.- Fuzzy Equivalence Relations.- Pattern Classification.- Similarity Relations.- References.- Fuzzy Graphs: Paths and Connectedness. Bridges and Cut Vertices. Forests and Trees. Trees and Cycles. A Characterization of Fuzzy Trees. (Fuzzy) Cut Sets. (Fuzzy Chords, (Fuzzy) Cotrees, and (Fuzzy) Twigs. (Fuzzy) 1-Chain with Boundary 0, (Fuzzy) Coboundary, and (Fuzzy) Cocycles. (Fuzzy) Cycle Set and (Fuzzy) Cocycle Set.- Fuzzy Line Graphs.- Fuzzy Interval Graphs. Fuzzy Intersection Graphs. Fuzzy Interval Graphs. The Fulkerson and Gross Characterization. The Gilmore and Hoffman Characterization.- Operations on Fuzzy Graphs: Cartesian Product and Composition. Union and Join.- On Fuzzy Tree Definition.- References.- Applications of Fuzzy Graphs: Clusters.- Cluster Analysis. Cohesiveness. Slicing in Fuzzy Graphs.- Application to Cluster Analysis.- Fuzzy Intersection Equations. Existence of Solutions.- Fuzzy Graphs in Database Theory. Representation of Dependency Structure r(X,Y) by Fuzzy Graphs.- A Description of Strengthening and Weakening Members of a Group. Connectedness Criteria. Inclusive Connectedness Categories. Exclusive Connectedness Categories.- An Application of Fuzzy Graphs to the Problem Concerning Group Structure. Connectedness of a Fuzzy Graph. Weakening and Strenghtening Points of a Fuzzy Directed Graph.- References.- Fuzzy Hypergraphs: Fuzzy Hypergraphs.- Fuzzy Transversals of Fuzzy Hypergraphs. Properties of Tr(H). Construction of H3.- Coloring of Fuzzy Hypergraphs. beta-degree Coloring Procedures. Chromatic Values of Fuzzy Colorings.- Intersecting Fuzzy Hypergraphs. Characterization of Strongly Intersecting Hypergraphs. Simply Ordered Intersecting Hypergraphs. H-dominant Transversals.- Hebbian Structures.- Additional Applications.- References.

Data Mining Through Fuzzy Social Network Analysis

In this paper, fuzzy theory has been applied to social network analysis (SNA). Social network analysis models meaningful relations that exist between entities as graph. These entities may be people, events, organizations, symbols in text, sounds in verbalizations, nations of the world and so on. However, the fuzzy graph can be very huge and thus the ability to arrive at meaningful conclusions in a timely fashion may be quite difficult. With this in mind, a method to consolidate the information content of the fuzzy graph is proposed. Since none of the existing fuzzy binary operations meet the requirements, a new fuzzy binary operation called consolidation operation is also introduced.

Cycles and cocycles of fuzzy graphs

Abstract In this paper we show that if the fuzzy graph (σ,μ) is a cycle, then it is a fuzzy cycle if and only if (σ,μ) is not a fuzzy tree. We also examine the relationship between fuzzy bridges and cycles. We introduce and examine the concepts of chords, twigs, 1-chains with boundary zero, cycle vectors, coboundary, and cocycles for fuzzy graphs. We show that although the set of cycle vectors, fuzzy cycle vectors, cocycles, and fuzzy cocycles do not necessarily form vector spaces over the field Z 2 of integers modulo 2, they nearly do. Thisallows us to introduce the concepts of (fuzzy) cycle rank and (fuzzy) cocycle rank for fuzzy graphs in a meaningful way.

Successor and source of (fuzzy) finite state machines and (fuzzy) directed graphs

A similarity between finite state machines and directed graphs can be seen from the natural way a directed graph can be associated with a finite state machine to describe the state transition of the finite state machine. Likewise, there is a similarity between fuzzy finite state machines and fuzzy directed graphs. As a matter of fact, all four of these concepts, together with that of information retrieval systems and fuzzy systems, share this similarity, namely that of the notion of successor. This paper gives an axiomatic treatment of the notion of successor in such a way that all of the above systems fall under this axiomatic approach.

Fuzzy generators and fuzzy direct sums of Abelian groups

Abstract We define the concept of a fuzzy generating set and describe the fuzzy subgroup which it generates. We introduce the notion of a minimal fuzzy generating set and we show that any fuzzy subgroup whose support is cyclic has a minimal fuzzy generating set. We show by examples that this result breaks down if the support of A is a direct sum of two or more cyclic groups. These examples also show that a fuzzy subgroup of a group G need not be a fuzzy direct sum of subgroups whose supports are cyclic even if G is a direct sum of cyclic groups. We show that every fuzzy subgroup is a fuzzy direct sum of p-primary fuzzy subgroups.

Perfect and precisely perfect fuzzy graphs

In this paper, we consider two different definitions of adjacency in fuzzy graph. Thus any graph theory concept that can be defined through adjacency can have two different variations in fuzzy graph theory. We demonstrate this fact through perfect graph and perfect graph theorem. Thus in the case of fuzzy graphs we have two independent concepts: perfect fuzzy graphs and precise perfect fuzzy graphs.

FUZZY MEALY MACHINES: HOMOMORPHISMS, ADMISSIBLE RELATIONS AND MINIMAL MACHINES

Homomorphisms and admissible relations of fuzzy Mealy machines are studied. Admissible relations play a role similar to normal subgroups in group theory. The kernel of a homomorphism is shown to be an admissible relation. Conversely, corresponding to an admissible relation, there exists a homomorphism. The fundamental theorem on homomorphisms; and the existence and uniqueness of minimal machines are also presented.

Ciset relational database for seamless integration of multi-source data

The aim of this paper is to introduce ciset relational database. The relational database lacks the ability to combine opposing and supporting evidences related to a fact. On the other hand, the ciset relational database has the capability to integrate both supporting and opposing evidences relating to the same fact in a seamless fashion. The ciset relational database is a generalization of the relational database theory. Given two relations A and B, one may convert them to corresponding cisets. Performing a ciset operation and converting the result back to a relation is equivalent to applying the relational operation on A and B directly. Similarly, if you perform a relational operation first and then convert them to ciset relation it has the same effect as converting to ciset and then performing the ciset operation. Concepts such as primary key, foreign key and data integrity rules are also presented.

Applications of Fuzzy Graphs

Let (V, μ, ρ) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” dis(x,y) between x and y as the length of the shortest strongest path between them. This “distance” is symmetric and is such that dis(x,x) = 0 since by our definition of a fuzzy graph, no path from x to x can have strength greater than μ(x), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let V = {u, v, x, y,z}, ρ(x, u) = ρ(u, v) = ρ(v, z) = 1 and ρ(x, y) = ρ(y, z) = 0.5. Here any path from x to y or from y to z has strength ≤ 1/2 since it must involve either edge (x,y) or edge (y, z). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from x to z, through u and v, that has length 3 and strength 1. Thus dis(x,z) = 3 > 1 + 1 = dis(x,y) + dis(y, z) in this case.

Fuzzy Mealy machines

Presents fundamental results on fuzzy Mealy machines. Unlike the classical Mealy machine which requires two functions, one to describe the next state and another to describe the output, a fuzzy Mealy machine requires only one fuzzy function to characterize completely the next state and the output produced. Apart from the obvious generalization that can be obtained from corresponding results on fuzzy finite state machines by introduction of an output associated with each transition, introduces the concept of an interval partition of [0, 1] and uses it to obtain more general results.