Document Type : Research Paper
Authors
1 COMSATS University Islamabad Abbottabad Campus
2 COMSATS University Islamabad, Abbottabad Campus.
Abstract
Over the years, traditional fuzzy graph models such as Intuitionistic Fuzzy Graphs (IFGs), Pythagorean Fuzzy Graphs (PFGs), and q-Rung Orthopair Fuzzy Graphs (q-ROFGs) have emerged as powerful tools for modeling uncertainty in complex networked structures. Despite their advancements, these models exhibit critical limitations, particularly in scenarios where the sum of membership and non-membership degrees exceeds one. For example, graph vertex data of the form {(v₁,(1,0.5)),(v₂,(1,1)),(v₃,(0.3,1))} or edge data of the form {(e₁,(1,1)),(e₂,(1,0.6)),(e₃,(0.3,1))} violate the constraint 0≤μ+ν≤1. Moreover, existing models often struggle to eliminate the inherent dependency between the membership and non-membership grades.
To overcome these limitations, we propose a novel framework called the q-Fractional Fuzzy Graph (q-FFG), which extends fuzzy graph theory, allows greater flexibility and independence between membership and non-membership grades. Unlike previous models, q-fractional fuzzy graphs can handle extreme degrees of uncertainty by letting both membership and non-membership values approach 1 independently, without violating consistency. We formally define the structure of q-fractional fuzzy graphs and some operations on these graphs. Additionally, we discuss degree and adjacency matrices of these graphs. These fundamental matrices serve as essential tools for understanding the structure and behavior of these graphs. Through illustrative examples, we highlight their practical applications and significance. The study presents an extensive review of the Laplacian matrices and their associated energies, uncovering essential information about the graph's spectral properties. These mathematical formulations are necessary for analyzing the stability and connectivity of q-fractional fuzzy graphs. Additionally, an algorithm based on Principal Component Analysis (PCA) technique is proposed which effectively reduces the dimensionality of data while preserving the essential features.
Keywords
Main Subjects
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