The aim of this paper is to introduce a new class of sets called nano preregular β-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 npβ-continuity is introduced.

In this article, we introduced few theorems related to the fractional integral image for product of q-analogue of general class polynomials with q- analogue of the generalized hypergeometric Functions. We used q-fractional -integral of Erdelyi-Kober type and generalized Weyl type fractional operators to prove our main results. The article concludes by indicating a wide range of results that can be obtained by using the relation between Erdelyi-Kober type and Riemann Liouville q-fractional integral, as well as, the relation between the generalized Weyl type and Weyl type q-fractional integrals.

In this paper, I introduce and study the properties of new class of contra continuous functions via the new sets called λB-closed sets and has as purpose to investigate some properties of contra-λB-continuous functions, contra-BRC-continuous functions, contra-Bg-continuous functions by using λB-open sets.

The concept of corona domination in graphs was introduced by G. Mahadevan et.al. recently. By the motivation of above parameter, in this paper we introduce another new parameter called complementary perfect corona domination in graphs. Let G = (V; E) be a graph. A dominating set S of a graph G is said to be a complementary perfect corona dominating set (CPCD { set) if each vertex in < S > is either a pendent vertex or a support vertex and < V − S > has a perfect matching. The minimum cardinality of a complementary perfect corona dominating set is called the complementary perfect corona domination number and is denoted by γCP C(G). In this paper we initiate a study of this parameter and found this number for some standard graphs.

An edge - magic total labeling on a (p, q) – graph G is a one - to - one map f from V(G) U E(G) onto the integers 1, 2, . . . , p + q, where p = |V(G)| and q = |E(G)|, with the property that given any edge xy, f(x) + f(y) + f(xy) = k for some constant k . The constant k is called the magic sum of f. Any graph G with an edge - magic total labeling is called edge - magic. In a communication network with p nodes and q links between nodes, applying edge - magic total labeling, we can assign unique address to the link from x to y as k - f(x) - f(y). Moreover a unique address f(x) to each node x, is available for messages from the system administrator. An edge - magic graph G with an edge - magic labeling f, is said to be super edge - magic if f(V(G)) = {1, 2, . . . ,p}. There are graphs that are not super edge - magic. This fact motivates the emergence of the concept of the super edge - magic deficiency of a graph. The super edge - magic deficiency of a graph G, is defined as either the minimum non - negative integer n such that G U nK 1 is super edge - magic; or + ∞, if there exists no such integer n. It is denoted by µ s (G). For the fan F n = P n + K 1, the graph obtained by joining each vertex in a path on n vertices to a single vertex, µ s (F n ) = 0, for 1 ≤ n ≤ 6; and µ s (F n ) = 1, for 7 ≤ n ≤ 11. In this paper, we have proved that µ s (F n ) = 2, for 12 ≤ n ≤ 16.

The concept of soft sets was introduced by Molodtsov in 1999, as a general mathematical tool to deal with uncertain objects. Later, in 2011, Shabir and Naz and Çağman et al., both initiated the study of soft topology and soft topological spaces independently. The geometric existence of N-topology was introduced by Lellis Thivagar et al., [4] in 2016, which is a nonempty set equipped with N-arbitrary topologies. In this paper, we attempt to extend the concept of Soft topological space to N- Soft topological spaces. Further, we characterize N-soft open, N-soft closed, N-soft neighborhood, and N-soft subspaces are introduced with suitable examples. It provides a base to research work in the field of N-soft topology and will help to establish a general framework for applications in practical fields.