An Introduction to Picture Fuzzy Graph and Its Application to Select Best Routes in an Airlines Network

An Introduction to Picture Fuzzy Graph and Its Application to Select Best Routes in an Airlines Network

S. Amanathulla, Madhumangal Pal
Copyright: © 2022 |Pages: 27
DOI: 10.4018/978-1-7998-7979-4.ch018
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Abstract

A picture fuzzy set is an extension of intuitionistic fuzzy set. In this set, a new parameter called neutral value of an object is incorporated. Based on this set, the picture fuzzy graph is defined and investigated by several scientists. Picture fuzzy graph is a new type of fuzzy graph, and it is used for solving many real life problems. In this chapter, the concept of picture fuzzy set is presented. It also introduced the picture fuzzy graph and investigated some useful properties of picture fuzzy graph. Some fundamental terms like degree of a vertex, order and size of picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, independent strong and independent weak picture fuzzy edge, bipartite picture fuzzy graph, complement of a picture fuzzy graph, path, strength of a path, connectedness, homomorphism, isomorphism, automorphism, Cartesian product, composition, etc. are defined and presented some properties. An application is provided to select best airline route among multiple paths.
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Basic Concept Of Fuzzy Graph

IFG is a generalization of fuzzy graph. The definition of IFG is given below:

Definition 1 An IFG is of the form G=(T,𝜎,𝜇) where 978-1-7998-7979-4.ch018.m01 and

  • (i)

    T={t0,t1,…,tn} such that 𝜎1: T→[0,1] and 𝜎2: T→[0,1], denote the degree of membership and non-membership of the node ti∈T respectively and 978-1-7998-7979-4.ch018.m02 for every ti∈T(i=1,2,…,n).

  • (ii)

    𝜇1: T×T→[0,1] and 𝜇2: T×T→[0,1], where 𝜇1(ti,tj) and 𝜇2(ti,tj) denote the the degree of membership and non-membership value of the edge (ti,tj) respectively such that

    978-1-7998-7979-4.ch018.m03
    ,

978-1-7998-7979-4.ch018.m04 for every (ti,tj).

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