The Strong Spectral Property and the Jacobian Method for Weighted Laplacian Matrices

Abstract

Strong matrix properties, roughly speaking, refer to generic conditions on a matrix such that its spectral perturbation and pattern perturbation interact nicely to cover a neighborhood in the ambient space. With a rich history, these strong properties originate from various fields, including the inverse eigenvalue problem, the sign pattern problem, and structural graph theory. In this paper, we introduce a new strong property, the strong spectral property for weighted Laplacian matrices (SSPWL), and establish the corresponding Supergraph and Bifurcation lemmas. Instead of the space of symmetric matrices, the SSPWL considers the ambient space spanned by all weighted Laplacian matrices. Moreover, we provide a detailed study comparing the Jacobian Method and some strong properties, leading to a full understanding between these two techniques used in different problems. Using these tools, we identify the potential boundaries of the spectral regions of weighted Laplacian matrices associated with connected graphs on $4$ vertices, extending the analysis from the previous work [S. M. Fallat, H. Gupta, and J. C.-H. Lin. Inverse eigenvalue problem for Laplacian matrices of a graph. SIAM J. Matrix Anal. Appl., 46:1866--1886, 2025]. In addition, this analysis can be used to identify the absolute algebraic connectivity of such small ordered graphs, and we establish the existence of strong weighted Laplacian matrices for several graph families.