Analysis of the effect of medicines over bacteria based on competition graphs with picture fuzzy environment
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Analysis of the effect of medicines over bacteria based on competition graphs with picture fuzzy environment Sankar Das1,2 · Ganesh Ghorai1 Received: 23 October 2019 / Revised: 23 March 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this study, the concept of picture fuzzy competition graph along with its two subclasses such as picture fuzzy k-competition graphs and p-competition picture fuzzy graphs are introduced. Picture fuzzy competition graph is one of the generalization of competition graph. Several properties of picture fuzzy competition graphs have been investigated. Some new types of picture fuzzy graphs have been introduced like picture fuzzy open neighborhood graph and picture fuzzy closed neighborhood graph. In addition, the relationship between picture fuzzy k-competition graph and picture fuzzy k-neighborhood graph are established. An application of picture fuzzy competition graph in medical science is presented. Keywords Picture fuzzy digraphs · Picture fuzzy competition graphs · Picture fuzzy k-competition graphs · p-competition picture fuzzy graphs · Picture fuzzy neighborhood graphs Mathematics Subject Classification 05C07 · 05C20 · 05C72 · 05E30 · 94C15
1 Introduction 1.1 Research background In 1968, Cohen (1968) first introduced the concept of competition graph (CG) while studying − → − → inflictions of graph theory in relation with an ecological problem. Let G = (V , B ) be the − → digraph corresponding to a food cycle . A vertex r ∈ V ( G ) indicates that a species in the
Communicated by Marcos Eduardo Valle.
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Ganesh Ghorai [email protected] Sankar Das [email protected]
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Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India
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Department of Mathematics, Kharagpur College, Kharagpur 721 305, India 0123456789().: V,-vol
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− →− → food cycle and an edge (r , x) ∈ B ( G ) indicate that r victims on the species x. The species r and s compete for a victim x if they have a common victim x. Due to this harmony, Cohen (1968) defined graphs representing the relationship of competition within the species of food − → − → cycle. An undirected graph C( G ) of G is the CG with V as vertex set and having an edge − → − → − → (r , s) in C( G ) if there is a vertex x ∈ V such that (r , x), (s, x) ∈ B in G . The concept of fuzzy set (FS) was first introduced by Zadeh (1965) in 1965. In 1986, Atanassov (1986) presented the idea of intuitionistic fuzzy set (IFS) as a generalization of FS to handle uncertainty and incomplete information in real life problems. But they may not be efficient to represent some special types of information. In such cases, a component namely ’neutrality’ is needed to represent the information completely. To overcome this issue, picture fuzzy set (PFS) was introduced by Cuong (2014) in 2014 as a generalization of IFS. The new idea of similarity measures for picture hesitant FSs with
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