Certain competition graphs based on picture fuzzy environment with applications
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Certain competition graphs based on picture fuzzy environment with applications Sankar Das1,2 • Ganesh Ghorai1
•
Madhumangal Pal1
Springer Nature B.V. 2020
Abstract In this paper, the notion of picture fuzzy competition graph along with its few generalizations such as m-step picture fuzzy competition graphs, picture fuzzy economic competition graphs and picture fuzzy competition hypergraphs are introduced. Some related picture fuzzy graphs including picture fuzzy m-step neighborhood graph, picture fuzzy mstep economic competition graph and picture fuzzy k-competition hypergraphs are introduced. Some properties of these graphs have been investigated. Finally, applications of mstep picture fuzzy competition graphs and picture fuzzy competition hypergraphs are presented in several fields such as in education system, ecosystem, business market and job competition. Keywords Picture fuzzy competition graphs m-step picture fuzzy competition graphs Picture fuzzy economic competition graphs Picture fuzzy hypergraphs Picture fuzzy competition hypergraphs
1 Introduction 1.1 Research background The notion of competition graph (CG) is formally introduced by Cohen (1968). In ecology, there are some problems of competition between species of food cycles those are modeled ! ! by the digraph G ¼ ðV; B Þ. These models are suitable to specify well defined nature of objects and especially species-victim relations. Nowadays, besides of ecosystem, CGs have many applications in other fields, such as coding and energy system, channel assignments, & Ganesh Ghorai [email protected] Sankar Das [email protected] Madhumangal Pal [email protected] 1
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India
2
Department of Mathematics, Kharagpur College, Kharagpur 721 305, India
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social interactions, modeling of complex economic, communications over a noisy channel ! ! ! etc. Let G ¼ ðV; B Þ be a digraph, which corresponds to a food cycle, a vertex r 2 Vð G Þ !! represent a species and arc ðr; qÞ 2 B ð G Þ indicate that r kills the species q. If two species ! ! r and s have a common victim q then they will compete for q. The CG Cð G Þ of G is an ! undirected graph with V as vertex set and having an edge (r, s) in Cð G Þ iff there is a vertex !! q 2 V such that ðr; qÞ; ðs; qÞ 2 B ð G Þ for any r; s 2 V, ðr 6¼ sÞ. The notion of fuzzy set (FS) was first introduced by Zadeh (1965) to discuss the uncertainty in several real life problems. It was found that one component is not sufficient to describe some special types of information. In this situations, a component namely nonmembership value is invited to illustrate the information properly and in addition to this new component (Atanassov 1986) defined intuitionistic fuzzy set (IFS). Later on, in some cases, another component namely ‘neutrality’ is needed to represent some information completely. To recover these scenarios, Cuong (2014)
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