The bipolar fuzzy set and bipolar soft set have inspired the development of a new framework called double-framed bipolar fuzzy soft sets (DFBFSSs). This structure represents positive and negative membership information through ordered pairs, enabling a balanced treatment of uncertainty, imprecision, and bi-directional information in complex decision-making scenarios. The fundamental concepts and operations of DFBFSSs are rigorously defined and analyzed. The double-framed formulation is symmetric: exchanging the frames preserves the structure of DFBFSSs. This symmetry enables balanced handling of opposing or complementary information. The key properties of the proposed set show improved handling of uncertainty over existing fuzzy and soft set models. In addition, a decision-making algorithm based on DFBFSSs is developed and applied to a real-world problem to validate the framework’s feasibility. Comparative analysis confirms the method’s robustness and advantages in uncertain, dual-information settings.

In this work, generalized Euler’s Φw-function of edge weighted graphs is defined which consists of the sum of the Euler’s φ-function of the weight of edges of a graph and we denote it by Φw(G) and the general form of Euler’s Φw-function of some standard edge weighted graphs is determined. Also, we define the divisor sum Tkw-function Tkw(G) of the graph G, which is counting the sum of the sum of the positive divisor σk-function for the weighted of edges of a graph G. It is determined a relation between generalized Euler’s Φw-function and generalized divisor sum Tkw-function of edge weighted graphs.

Let G = (V, E) be a simple connected undirected graph. In this paper, we define generalized the Liouville’s and Möbius functions of a graph G which are the sum of Liouville λ and Möbius µ functions of the degree of the vertices of a graph denoted by Λ(G) = ∑v∈V(G)λ(deg(v)) and M(G) = ∑v∈V(G)µ(deg(v)), respectively. We also determine the Liouville’s and Möbius functions of some standard graphs as well as determining the relationships between the two functions with their proofs. The sum of generalized the Liouville and Möbius functions extending over the divisor d of degree of vertices of graphs is also given

In this paper, we apply the notion of Pp-open sets in topological spaces to present and study a new class of functions called contra Pp-continuous functions which lies between classes of contra θ-continuous functions and contra-precontinuous functions. It is shown that contra Pp-continuous is weaker than contra θ-continuous, but it is stronger than contra-precontinuous and weakly Pp-continuous. Furthermore, we obtain basic properties and preservation theorems of contra Pp-continuity.

In this paper we introduce a new class of sets, called Pp-open sets, also using this set, we define and investigate some properties of the concept of Pp-continuity. In particular, Pp-open sets and Pp-continuity are defined to extend known results for preopen sets and pre-continuity.