The object of this paper is to develop the transformations of the k-hypergeometric function of the type , which has been converted in to know results. Further we also establish the k-gauss theorem and k-gauss hypergeometric equation with its solutions

This research aims to find the fractional order derivative of generalized Mittag-Leffler Factorial Function (MLFF). The Generalized Mittag-Leffler factorial function is obtained by replacing polynomials into polynomial factorials in the expression of Mittag-Leffler Function. The Special case of this MLFF is the Extorial function. This function is used to solve the certain type of fractional difference equation and it has some applications in the field of Discrete Fractional Calculus.

Designing of interconnection network plays an important in modelling large scale parallel architectures. Some notable interconnection networks are cayley graphs. There has been a spurt of research on cayley graphs as it has gained importance for the past few years that are suitable for designing interconnection networks. Motivated by this, we would like to investigate our research on slope number and domatic number on cayley graphs. Here, the slope number is about minimizing the slopes and domatic number is about partitioning the vertices into dominating sets. The main objective of this paper is to present characterization theorem on certain classes of cayley graphs. We consider cayley graphs such as complete graph, crown graph and circulant graph for analysis and the paper is divided into three sections. In the first section, we examine the slope number for these graphs. In the second section, we observed results on domatic number. Finally we provide relationship between slope number and domatic number based on the characterization that satisfies equality condition.

In this paper, we study and establish some interesting results of J-ideals in posets. Characterization of prime ideals to be J-ideals and the equivalent conditions for an ideal to be J-ideal of P are obtained. Finally the separation theorem for J-ideal is also discussed.

The aim of this paper is to introduce a new class of sets called nano preregular ?-open sets in nano topological spaces. Some properties and characterizations of the said type of sets are investigated. Also, a new class of nano continuity called np?- continuity is introduced.

The aim of this chapter gives more valuable information about many Fractal Graphs. It analyses the structure, implementation of vertices, Edges and angles of Fractal Graph for all Iteration. It finds out the implementation of Vertices and Edges for all Iteration follows the Constant Formulae in the Fractal Graphs. This chapter finds that in which common formulae is applied for the implementation of vertices and Edges in the Fractal Graph like Von Koch Curve (which has even number of edges in each iteration), Sierpenski Arrow Head Curve (which has odd number of edges in each iteration). Matching is one of the very important and more scope topic in Graph Theory. Calculation of Maximum Matching is one of the major evaluations in this chapter. In this chapter separates Fractal Graph into two parts like Open Walk Fractal Graph and Closed Walk Fractal Graph. This paper finds the new common formulae for calculating Matching cardinality which depends on the total number of vertices and total number of edges in the corresponding Iteration of the given Fractal Graph. Here check and examine whether the derived formulae is suitable in some of the Open Path Fractal Graph like Von Koch Curve and Sierpenski Arrow Head Curve. This paper conclude that this derived formulae can be implemented to all Open Path Fractal Graphs. Calculation of Maximum Matching can be determined by using Iterative Methods.

In this paper, we have developed a new method through statistical third quartile process to find out the initial basic feasible solution of the given transportation problem and compare it with Least Cost Method / minimum cell cost method as well as North � West Corner method and shown that the statistical third quartile method is best as compare to least cost and North � West Corner method.

This research aims to find the fractional order derivative of Extorial Function. The Extorial function is obtained by replacing polynomials into polynomial factorials in the exponential function. This function is used to solve the fractional order difference equations by using the inverse of higher order difference operator.