### A Fiedler-like theory for the perturbed Laplacian

The perturbed Laplacian matrix of a graph $G$ is defined as ${L}^{\phantom{\rule{-8.33328pt}{0ex}}D}=D-A$, where $D$ is any diagonal matrix and $A$ is a weighted adjacency matrix of $G$. We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore, we use...