International Journal of Mathematics And its Applications Volume 5, Issue 4–F (2017), 915–922. ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Homeomorphism of Fuzzy Topological Graph Research Article P. Jayalakshmi 1 and N. Ramya Priya 1* 1 Department of Mathematics, Sri GVG Visalakshi College for Women (Autonomous), Udumalpet, Tamilnadu, India. Abstract: This paper provides solid theoretical foundation for studying Fuzzy Topological Graph theory. The concepts such as subdivision, homeomorphism and connectedness of a fuzzy topological graph is established. In addition to this an attempt is made to study about planar and minimal non planar fuzzy topological graphs. These concepts are illustrated through examples. Keywords: Homeomorphism, Fuzzy Topological Graph, Subdivision, Planar Fuzzy Topological Graph, Minimal Non Planar Fuzzy Topological Graph, Connectedness of Fuzzy Topological Graph. c JS Publication. 1. Introduction and Preliminaries L.A.Zadeh introduced Fuzzy set theory in 1965 describing fuzziness mathematically for the first time. The fuzzy graph theory was introduced by A.Rosenfeld using fuzzy relation representing the relationship between the objects by precisely indicating the level of the relationship between the objects of the given set. The notion of fuzzy topology was introduced by C.L.Chang in 1968. It is extension of the concepts of ordinary topological space where the family of all the fuzzy sets on the universe X takes I=[0,1] as the range by I X . Substituting inclusive relation by the order relation in I X , a topological structure is introduced naturally into I X . Topological graph theory deals with ways to represents the geometric realization of graphs. In early 1987, the frontiers of topological graph theory are advancing in numerous different directions.This is the background to introduce the new concept fuzzy topological graph and some of its properties are discussed. Definition 1.1 (Fuzzy graph [1]). A fuzzy graph is a pair G :(σ, μ) where σ is a fuzzy subset of S, μ is a symmetric fuzzy relation on σ. The elements of S are called the nodes or vertices of G and the pair of vertices edges in G. Definition 1.2 (Path [2]). A path P in a fuzzy graph G :(σ, μ) is a sequence of distinct nodes v0,v1,v2,...,vn such that μ(vi-1,vi ) > 0, 1 <i<n. Here n is called the length of the path. The consecutive pairs (vi-1,vi ) are called arcs of the path. Definition 1.3 (Fuzzy connectedness [1]). If u,v are nodes in G and if they are connected by means of a path the strength of that path is defined as n ∧ i=1 μ (vi-1,vi ). (i.e.) It is the strength of the weakest arc. If u,v are connected by means of path of length ‘k’ then μ k (u, v) is defined as μ k (u, v) = sup {μ (u, v1) ∧ μ (v1,v2) ∧ μ (v2,v3) ∧ ... ∧ μ (v k-1 ,v) /u, v1, v2...v k-1 ,v ∈ S}. If u, v ∈ S the strength of connectedness between u and v is μ ∞ (u, v) = sup{μ k (u, v) /k =1, 2, 3, ..}. * E-mail: [email protected]
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Homeomorphism of Fuzzy Topological Graphijmaa.in/v5n4-f/915-922.pdf · Topological graph theory deals with ways to represents the geometric realization of graphs. In early 1987, the
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International Journal of Mathematics And its Applications
Volume 5, Issue 4–F (2017), 915–922.
ISSN: 2347-1557
Available Online: http://ijmaa.in/
Applications•ISSN:234
7-15
57•In
ternationalJo
urna
l of MathematicsAnd
its
International Journal of Mathematics And its Applications
Homeomorphism of Fuzzy Topological Graph
Research Article
P. Jayalakshmi1 and N. Ramya Priya1∗
1 Department of Mathematics, Sri GVG Visalakshi College for Women (Autonomous), Udumalpet, Tamilnadu, India.
Abstract: This paper provides solid theoretical foundation for studying Fuzzy Topological Graph theory. The concepts such assubdivision, homeomorphism and connectedness of a fuzzy topological graph is established. In addition to this an attempt
is made to study about planar and minimal non planar fuzzy topological graphs. These concepts are illustrated through
examples.
Keywords: Homeomorphism, Fuzzy Topological Graph, Subdivision, Planar Fuzzy Topological Graph, Minimal Non Planar FuzzyTopological Graph, Connectedness of Fuzzy Topological Graph.