  The smallest nonzero eigenvalue of the normalized Laplacian matrix of a graph
has been extensively studied and shown to have many connections to properties
of the graph. We here study a generalization of this eigenvalue, denoted
$\lambda(G, X)$, introduced by Mendel and Naor in 2010, obtained by embedding
the vertices of the graph $G$ into a metric space $X$. We consider general
bounds on $\lambda(G, X)$ and $\lambda(G, H)$, where $H$ is a graph under the
standard distance metric, generalizing some existing results for the standard
eigenvalue. We consider how $\lambda(G, H)$ is affected by changes to $G$ or
$H$, and show $\lambda(G, H)$ is not monotone in either $G$ or $H$.
