  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.
