Fuzzy soft graphs are effective mathematical tools that are used to model the vagueness of the real world. A fuzzy soft graph is a fusion of the fuzzy soft set and the graph model and is widely used across different fields. In this current research, the concept of picture fuzzy soft graphs is presented by combining the theory of picture soft sets with graphs. The introduction of this new picture fuzzy soft graphs is an emerging concept that can be rather developed into various graph theoretical concepts. Since soft sets are most usable in real-life applications, the newly combined concepts of the picture and fuzzy soft sets will lead to many possible applications in the fuzzy set theoretical area by adding extra fuzziness in analyzing. The notions of picture soft graphs, strong and complete picture soft fuzzy graphs, a few types of product picture fuzzy soft graphs, and regular, totally regular picture fuzzy soft graphs are discussed and validated using real-world scenarios. In addition, an application of decision-making for medical diagnosis in the current COVID scenario using the picture fuzzy soft graph has been illustrated.
The concept of interval valued pentapartitioned neutrosophic set is the extension of interval-valued neutrosophic set, quadripartitioned neutrosophic set, interval valued quadripartitioned neutrosophic set and pentapartitioned neutrosophic set. The powerful mathematical tool known as the interval valued pentapartitioned neutrosophic set divides indeterminacy into three separate components: unknown, contradiction, and ignorance. There are several applications for graph theory in everyday life, and it is a rapidly growing topic. The concept of an interval valued pentapartitioned neutrosophic set is used in graph theory. A decision-making method multicriteria (MCDM) is proposed by using the developed Interval valued Pentapartitioned Neutrosophic set with a numerical illustration. In this paper, as an extension of interval valued neutrosophic graph theory, we introduce the notions of Interval-Valued Pentapartitioned Neutrosophic Graph (IVPPN-graph) with degree, size, and order of an IVPPN-graph.
The main focus of this paper is to develop certain types of fundamental theorems using q, q(α), and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q(α), and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q(α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q(α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q(α) difference equations.
<abstract><p>Pythagorean neutrosophic set is an extension of a neutrosophic set which represents incomplete, uncertain and imprecise details. Pythagorean neutrosophic graphs (PNG) are more flexible than fuzzy, intuitionistic, and neutrosophic models. PNG are similar in structure to fuzzy graphs but the fuzziness is more resilient when compared with other fuzzy models. In this article, regular Pythagorean neutrosophic graphs are studied, where for each element the membership $ (\mathfrak{M}) $, and non-membership $ (\mathfrak{NM}) $ are dependent and indeterminacy $ (\mathfrak{I}) $ is independently assigned. The new ideas of regular, full edge regular, edge regular, and partially edge regular Pythagorean Neutrosophic graphs are introduced and their properties are investigated. A new MCDM method has been introduced using the Pythagorean neutrosophic graphs and an illustrative example is given by applying the proposed MCDM method.</p></abstract>
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