On the Smallest Eigenvalue of Grounded Laplacian Matrices

TL;DR

This paper establishes bounds and asymptotic behaviors for the smallest eigenvalue of grounded Laplacian matrices, with implications for consensus dynamics in networks.

Contribution

It provides new bounds and characterizations for the smallest eigenvalue of grounded Laplacians, especially for random and regular graphs, linking spectral properties to graph structure.

Findings

For Erdős-Rényi graphs, the smallest eigenvalue approaches |S|p when p is large.

For random d-regular graphs, the smallest eigenvalue is Θ(d/n).

Bounds relate eigenvector component ratios to graph properties.

Abstract

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the 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 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 $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes.