AUTHOR COPY Journal of Intelligent & Fuzzy Systems 37 (2019) 2869–2885 DOI:10.3233/JIFS-190036 IOS Press 2869 Single-valued neutrosophic directed (Hyper)graphs and applications in networks Mohammad Hamidi a,∗ and Florentin Smarandache b a Department of Math., Faculty of Mathematics, Payame Noor University, Tehran, Iran b Department of Math., University of New Mexico, Gallup, NM, USA Abstract. This paper considers networks as wireless sensor (hyper)networks and social (hyper)networks by single–valued neutrosophic (directed)(hyper)graphs.The notion of single–valued neutrosophic hypergraphs are extended to single–valued neutrosophic directed hypergraphs and conversely. We derived single–valued neutrosophic digraphs from single– valued neutrosophic directed hypergraphs via a positive equivalence relation. It tries to use single–valued neutrosophic directed hypergraphs and positive equivalence relation to create the sensor clusters and to access to cluster heads in wire- less sensor (hyper)networks. Finally, the concept of α-derivable single–valued neutrosophic digraph is considered as the energy-efficient protocol of wireless sensor networks and is applied this concept as a tool in wireless sensor (hyper)networks. Keywords: Single–valued neutrosophic directed (graphs)hypergraphs, positive equivalence relation, α-(semiself- self)derivable single–valued neutrosophic digraph, SN, WSN 1. Introduction As a generalization of the classical set theory, fuzzy set theory was introduced by Zadeh [33] to deal with uncertainties. Fuzzy set theory is playing an impor- tant role in modeling and controlling unsure systems in nature, society and industry. Fuzzy set theory also plays a vital role in phenomena which is not easily characterized by classical set theory. Smarandache proposed the idea of neutrosophic sets and mingled thee component logic, non-standard analysis, and philosophy, in 1998 [26, 27]. Smarandache [26] and Wang et al. presented the notion of single-valued neutrosophic sets in real life problems more con- veniently [32]. A single-valued neutrosophic set has three components: truth membership degree, indeter- ∗ Corresponding author. Mohammad Hamidi, Department of Math., Faculty of Mathematics, Payame Noor University, Tehran, Iran. E-mail: m.hamidi@pnu.ac.ir. minacy membership degree and falsity membership degree. These three components of a single-valued neutrosophic set are not dependent and their val- ues are contained in the standard unit interval [0, 1]. Single-valued neutrosophic sets have been a new hot research topic and many researchers have addressed this issue. Majumdar and Samanta studied similarity and entropy of single-valued neutrosophic sets [16]. Smarandache [28, 29] have defined four main categories of neutrosophic graphs, two based on literal indeterminacy (I ), whose name were; I – edge neutrosophic graph and I –vertex neutrosophic graph, these concepts have been deeply studied and have gained popularity among the researchers due to their applications in real world problems [10, 30]. Akram et al. defined the concepts of single-valued neutrosophic hypergraph, line graph of single- valued neutrosophic hypergraph, dual single-valued neutrosophic hypergraph, transversal single-valued ISSN 1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved