Bounds on Geometric Eigenvalues of Graphs

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 $H$.