The purpose of this paper is to introduce a new structure called primal. Primal is the dual structure of grill. Like ideal, the dual of filter, this new structure also generates a new topology named primal topology. We introduce a new operator using primal, which satisfies Kuratowski closure axioms. Mainly, we prove that primal topology is finer than the topology of a primal topological space. Also, we provide the structure of the base of primal topology and prove other fundamental results related to this new structure. Furthermore, we not only discuss some of this new structure’s properties but also enrich it with many examples.

Recently, Acharjee, Özkoc and Issaka introduced a new structure in topology named primal. Primal is the dual structure of grill. The main purpose of this paper is to introduce an operator using primal and obtain some of its fundamental properties. Also, we define the notion of topology suitable for a primal. We not only obtain some characterizations of this new notion, but also investigate many properties.

Proximity space is one of the common topics in mathematics, computer science, and pattern recognition. Recently, Acharjee et al. introduced a new structure named primal in mathematics. Thus, the main purpose of this paper is to introduce and study primal-proximity spaces. Also, we define two new operators via primal proximity spaces and investigate some of their fundamental properties. In addition, we obtain a new topology, which is weaker than old one, via these new operators. Moreover, we not only discuss some of their properties but also enrich with some examples.

In this paper we introduce a new class of generalized closed sets in bitopological space using local function, two extension operators and semi-open sets. We have also investigated some properties in subspace bitopology defining kernel and image.

Topological indices describe mathematical invariants of molecules in mathematical chemistry. M-polynomials of chemical graph theory have freedom about the nature of molecular graphs and they play a role as another topological invariant. Social networks can be both cyclic and acyclic in nature. We develop a novel application of M-polynomials, the $(m,n,r)$ -agent recruitment graph where $n>1$ , to study the relationship between the Dunbar graphs of social networks and the small-world phenomenon. We show that the small-world effects are only possible if everyone uses the full range of their network when selecting steps in the small-world chain. Topological indices may provide valuable insights into the structure and dynamics of social network graphs because they incorporate an important element of the dynamical transitivity of such graphs.