SPECTRAL BOUNDS FOR FUZZY GRAPH LAPLACIANS UNDER INTERVAL UNCERTAINTY

Abstract

We study interval-valued fuzzy weighted graphs whose edge memberships are known only through closed intervals and investigate how this uncertainty propagates to the Laplacian spectrum. For every admissible realization we define the fuzzy weighted Laplacian and prove that the edgeto-Laplacian map is monotone in the Loewner order. Consequently the k-th Laplacian eigenvalue is confined to the exact interval [λk(L−), λk(L+)] generated by the lower and upper endpoint graphs. We derive robust connectivity criteria, midpoint perturbation bounds of Weyl type, Gershgorin-type global enclosures, and explicit formulas for homogeneous interval uncertainty on paths, cycles, stars, and complete graphs. Numerical examples show that exact spectral widths can be substantially smaller than generic perturbation radii. The results place interval-valued fuzzy graph Laplacians into a rigorous spectral-uncertainty framework.