Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.

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Recently, the concept of a soft rough fuzzy covering (briefly, SRFC) by means of soft neighborhoods was defined and their properties were studied by Zhan’s model. As a generalization of Zhan’s method and in order to increase the lower approximation and decrease the upper approximation, the present work aims to define the complementary soft neighborhood and hence three types of soft rough fuzzy covering models (briefly, 1-SRFC, 2-SRFC, and 3-SRFC) are proposed. We discuss their axiomatic properties. According to these results, we investigate three types of fuzzy soft measure degrees (briefly, 1-SMD, 2-SMD, and 3-SMD). Also, three kinds of ψ -soft rough fuzzy coverings (briefly, 1- ψ -SRFC, 2- ψ -SRFC, and 3- ψ -SRFC) and three kinds of D -soft rough fuzzy coverings (briefly, 1- D -SRFC, 2- D -SRFC, and 3- D -SRFC) are discussed and some of their properties are studied. Finally, the relationships among these three models and Zhan’s model are presented.

In this paper, we present a relation on graphs that induces new types of topological structures to the graph and then study some of the properties of this graph. Also, we investigate an algorithm to generate the topological structures from different graphs. Finally, some applications in medicine and geographs will be given, and we verify our results in the real life. INTRODUCTION AND PRELIMINARIESGraph theory 1-4 has recently emerged as a subject in its own right as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology, and genetics. A graph G is a pair (V, E), where V is nonempty set called vertices or nodes and E is 2-element subsets of V called edges or links. The number of vertices in a graph G is the order of G, and the number of edges is the size of G. An edge joining a vertex to itself is called a loop. Two or more edges that join the same pair of distinct vertices are called parallel edges. Let G = (V(G), E(G)) be a graph; we call H a subgraph ofThe eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex farthest from v in G, while the radius rad(G) is the smallest eccentricity among the vertices of G. The notions of closure operator is very useful tool in several sections of mathematics, as an example, in algebra, 5,6 topology, 7,8 and computer science theory, 9 the connection between graph theory and different subjects, as in structural analysis, 10 medicine 11 and physics. 12 Topology is the science that deals with the properties of things that does not depend on the dimension, which means that it allows increases and decreases, but without cutting on things. If X is a nonempty set, a collection of subsets of X is said to be a topology on X, and if the following condition holds X and belongs to , the finite intersection of any 2 sets in belongs to and the union of any number of sets in belongs to . 13 The term topology is also used to refer to a structure imposed upon a set X, a structure that essentially "characterizes" the set X as a topological space by taking proper care of properties such as convergence, connectedness, and continuity, upon transformation. Every element in topology is called an open set, its complement is a closed set. The closure of a subset A (briefly, Cl(A)) is the smallest closed set that contains A . The interior of a subset B (briefly, int(B)) is the greatest open set that is contained in B. The main contribution of the work is that we provide a new definition of a relation to extract a topology from any graph and study some properties. Throughout the paper, we start with the application of abstract topological graph theory. Some ideas in terms of concepts in topological graph theory, which is a branch of mathematics, and in many real-life applications will be investigated. We give an algorithm to generate some topological structural in graphs. Each topological structure on graphs is a topological space. Some properties on closure and interior operators for Math Meth Appl

As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzy β -neighborhoods and thus three kinds of covering-based multigranulation ( ℐ , T )-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ( ℐ , T )-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.