Permanent fuzzy graphs with application in decision making

Difference Square Mean Fuzzy Labelling of Graphs Constructed from Path Graph

Computer science applications make extensive use of graph theory. Information mining, picture classification, grouping images, photo capture, and connectivity are particularly important study areas in computer science. Fuzzy labelling models provide better accuracy, adaptability, and interoperability to the framework than conventional fuzzy models. They are used in many areas of basic mathematics, including traditional computer science and physical scientific study. A common problem in mathematics is the distance properties in a graph. In fuzzy mean labelling graphs, paths have some exciting applications. Using lengths to securely identify the vertex in an organization is one such application. As the outcome, in this study, we explore four paths that are DSMFLs in graphs: comb graph, double comb, path-attached pendant vertex, and path-attached two pendant vertices. In fuzzy labelling graphs, there exist several DSMFLs. Whenever the force of connection between each pair of vertices G matches the value of the membership of the edges, G appears self-sufficient in relation to the DSMF. Furthermore, it is established that any interconnected fuzzy labelling graph is both a path graph and a comb graph.

Connectivity Concepts in m-Polar Fuzzy Incidence Graphs

This paper introduces and investigates the connectivity properties with [Formula: see text]-PFIG, a novel extension of fuzzy incidence graphs characterized by multiple membership values corresponding to different polarities. The study systematically defines and explores key connectivity concepts, including paths, cycles, and connectedness, under the [Formula: see text]-polar fuzzy framework. The authors present theoretical findings that expand traditional graph connectivity concepts to a more generalized context, offering a deeper understanding of the behavior and structure of [Formula: see text]-PFIG. This study introduces an innovative method for connectivity concept and incidence graph within [Formula: see text]-polar fuzzy graphs ([Formula: see text]-PFG). Several properties, along with detailed descriptions are thoroughly investigated. The proposed approach accounts for the various merits of each member in a railway network system by considering individual merits (self-weight) in the nature.

Partition-based construction and stability analysis of Euler graphs using vertex strength

The ‘divide and conquer’ paradigm proves to be one of the most frequently used techniques for dealing with the complexities of graph-related problems. Therefore, it is of great importance to measure the tendency of a vertex to be critical and its susceptibility in a graph. The criticality of a vertex is often analysed in terms of its strength. Removing a highly critical vertex from a graph modelling a network may introduce vulnerability into the system represented by the graph. Minimizing the vulnerability of such a network without affecting its fundamental structure, thereby improving the stability of the graph, is the primary objective of the article. To achieve this, certain properties of Euler graphs are analysed in terms of vertex strength, and a method is presented for determining all possible constructions of Euler graphs corresponding to different integer partitions. The parts of a partition represent the vertex strengths, and their sum corresponds to the total vertex strength of the graph. Various connectivity indices are employed to validate the proposed constructions. Furthermore, their interrelationships and potential real-life applications are also discussed. It is evident from the constructions that they may play a vital role in developing network deception technology to protect digital assets, as each partition of the network generates a distinct network.

Structural Analysis of Ladder Corona Graphs via Intuitionistic Fuzzy Cube Difference Labeling

Intuitionistic fuzzy graph labeling is a useful approach for describing uncertainty based on membership (MS) and non-membership (NMS) values. This work studies Intuitionistic Fuzzy Cube Difference Labeling (IFCDL) on ladder-based graphs and corona products. Explicit labeling techniques are designed for ladder graphs, open ladder graphs, slanting ladder graphs, circular ladder graphs, and Mobius ladder graphs. It is demonstrated that these graphs permit intuitionistic fuzzy cube difference graphs (IFCDGs) by assigning distinct MS and NMS values within the range of [0, 1]. The suggested framework is adaptable to different graph products and complicated network models, with potential applications in decision-making, communication networks, and uncertainty-based systems.

Advanced Domination Concepts In Product Bipolar Fuzzy Graphs: Theory, Operations, And Applications

Bipolar fuzzy graphs (BFGs) extend classical fuzzy graph theory by in-corporating both positive and negative membership degrees, enabling the representation of dual-aspect uncertainty in complex systems. Product bipo-lar fuzzy graphs (PBfGs) provide a refined framework for modeling interde-pendent relationships where edge strength is determined muftipficatirefy by vertex attributes, rather than by the classical minimum or maximum opera-tors. This paper formulates and investigates four key advanced domination variants within the strict product-based constraints of PBfGs: secure dom-ination, which ensures network resilience under node failure; 2-domination, which guarantees redundant coverage for fault tolerance; connected perfect domination, which enforces structural connectivity and exactness of cover-age; and tiiùofe edge domination, which extends control mechanisms to edge-based network architectures. We establish theoretical foundations, prove characterization theorems and bounds, examine operational properties un-der graph products (Cartesian product, composition, union, and join), and demonstrate a real-world application to secure transit network design.

Characterization of inverse fuzzy mixed graphs in terms of cycles with application