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Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

In this paper, we apply the notion of soft rough neutrosophic sets to graph theory. We develop certain new concepts, including soft rough neutrosophic graphs, soft rough neutrosophic influence graphs, soft rough neutrosophic influence cycles and soft rough neutrosophic influence trees. We illustrate these concepts with examples, and investigate some of their properties. We solve the decision-making problem by using our proposed algorithm. Mathematics 2018, 6, 125 2 of 37 Kaufmann [22]. Later, Rosenfeld [23] considered fuzzy graphs and obtained analogs of several graph theoretical concepts. Mordeson and Peng [24] defined some operations on fuzzy graphs. Mathew and Sunitha [25,26] presented some new concepts on fuzzy graphs. Gani et al. [27-30] discussed several concepts, including size, order, degree, regularity and edge regularity in fuzzy graphs and intuitionistic fuzzy graphs. Parvathi and Karunambigai [31] described some operation on intuitionistic fuzzy graph. Recently, Akram et al. [32][33][34][35][36] has introduced several extensions of fuzzy graphs with applications. Denish [37] considered the idea of fuzzy incidence graph. Fuzzy incidence graphs were further studied in [38,39]. Due to the limitation of humans knowledge to understand the complex problems, it is very difficult to apply only a single type of uncertainty method to deal with such problems. Therefore, it is necessary to develop hybrid models by incorporating the advantages of many other different mathematical models dealing uncertainty. Recently, new hybrid models, including rough fuzzy graphs [40,41], fuzzy rough graphs [42], intuitionistic fuzzy rough graphs [43,44], rough neutrosophic graphs [45] and neutrosophic soft rough graphs [46] are constructed. For other notations and definitions, the readers are refereed to [47][48][49][50][51]. In this paper, we apply the notion of soft rough neutrosophic sets to graph theory. We develop certain new concepts, including soft rough neutrosophic graphs, soft rough neutrosophic influence graphs, soft rough neutrosophic influence cycles and soft rough neutrosophic influence trees. We illustrate these concepts with examples, and investigate some of their properties. We solve decision-making problem by using our proposed algorithm. This paper is organized as follows. In Section 2, some definitions and some properties of soft rough neutrosophic graphs are given. In Section 3, soft rough neutrosophic influence graphs, soft rough neutrosophic influence cycles and soft rough neutrosophic influence trees are discussed. In Section 4, an application is presented. Finally, we conclude our contribution with a summary in Section 5 and an outlook for the further research.

Furthermore, in this work, we derive some algebraic relations on the Pell formy 2 including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equationWe generalized all the results of the papers [2], [9], [26] and [37].