On the Smallest Eigenvalue of Grounded Laplacian Matrices

Abstract

We provide bounds on the smallest eigenvalue of grounded Laplacian matrices (which are obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between our upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for weighted Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $\mu_{w}\vert S\vert p$ , where $\mu_{w}$ is the mean edge weight. We also show that for weighted random $d$ -regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(1/n)$ , where $n$ is the number of nodes in the network. Our bounds have applications to the study of the convergence rate in consensus dynamics with stubborn or leader nodes.