Spectral Analysis of Bipolar Fuzzy Graphs through the Lens of Bipolar Fuzzy Laplacian

In this paper, a spectral framework for bipolar fuzzy graphs is developed to analyze networks involving both positive and negative relationships. A bipolar fuzzy Laplacian matrix is constructed by incorporating the positive and negative edge membership information into the corresponding degree and adjacency matrices. We investigate fundamental spectral properties of the proposed matrix, including positive semidefiniteness, the multiplicity of the zero eigenvalue, algebraic connectivity, and Laplacian spread. In addition, we establish a spectral stability result to quantify the effect of perturbations in bipolar membership values on the Laplacian spectrum. The proposed framework provides a unified spectral approach for measuring the structural characteristics of bipolar fuzzy networks and has potential applications in social and trust networks, decision-making, biological interaction networks, risk analysis, and supply-chain systems.