A Mathematical Framework for Fuzzy Graphs in Structural Analysis of Complex Networks: Centrality, Connectivity and Spectral Properties
Abstract: Complex networks frequently contain relationships whose existence or intensity is not known with certainty. Classical simple graphs force such relations into binary states, while weighted graphs usually treat weights as fixed numerical strengths. Fuzzy graph theory provides a natural mathematical language in which an edge membership µij ∈ [0, 1] expresses the degree to which a relation is supported. This paper develops a unifed mathematical framework connecting three structural viewpoints of a fuzzy network: centrality, connectivity and spectrum. Starting from a fnite undirected fuzzy graph G = (V, σ, µ), we construct the fuzzy adjacency matrix A = [µij], degree matrix D, Laplacian L = D-A, and the corresponding spectral parameters. A new spectral-fuzzy centrality index is proposed by combining normalized fuzzy degree with normalized two-step in?uence. The index remains in [0, 1], is invariant under vertex relabeling, is monotone under strengthening of incident memberships under a fixed normalization bound, and attains its maximal value on regular fuzzy networks under the proposed normalization. We further establish relations between the proposed centrality and classical weighted-network quantities, derive spectral connectivity bounds, and formulate an α-cut interpretation that links fuzzy structural analysis to a family of crisp realizations. An illustrative six-node network is analyzed exactly through its adjacency matrix, degree sequence, Laplacian spectrum and proposed centrality values. The framework is intended as a mathematically transparent bridge between fuzzy graph theory and complex-network analysis, with potential applications to uncertain social, communication, biological and renewable-energy networks.
Keywords: Fuzzy graph; complex network; fuzzy centrality; Laplacian spectrum; algebraic connectivity; spectral radius; α-cut; uncertain network; graph energy; structural robustness
How to Cite?: Dr. Guddu Kumar, Simran Kumari, "A Mathematical Framework for Fuzzy Graphs in Structural Analysis of Complex Networks: Centrality, Connectivity and Spectral Properties", Volume 15 Issue 10, October 2026, International Journal of Science and Research (IJSR), Pages: 697-707, https://www.ijsr.net/getabstract.php?paperid=SR261010160104, DOI: https://dx.doi.org/10.21275/SR261010160104