which can be bounded using $\lambda_2$ as follows: $$\label{eq:diambound}
  \frac{4}{N\lambda_2} \le \text{diam}{\left(H\right)} \le 2\ceil[\Big]{\frac{\Delta+\lambda_2}{4\lambda_2}\ln{\left(N-1\right)}},$$ where $\Delta$ is the maximum degree of $H$. It can be seen that a larger value for $\lambda_2$ will lead to a smaller graph diameter. We also have the following asymptotic bound on the mean distance between nodes, $\bar{\rho}(H)$: $$\label{eq:mdbound}
  \frac{2}{(N-1)\lambda_2(H)} + \frac{1}{2} \lesssim \bar{\rho}(H) \lesssim \ceil[\Big]{\frac{\Delta+\lambda_2}{4\lambda_2}\ln{\left(N-1\right)}}.$$ Another important diagnostic of a network is given by the *Cheeger constant* $h(H)$ [@Chung1997], also called the isoperimetric number or the graph conductance. This graph invariant is a measure of how difficult the graph is to disconnect by cutting edges. For a connected graph, this number is always positive. As benchmark values, the complete graph $K_N$ has Cheeger constant $N/2$ while a cycle graph $C_N$ has Cheeger constant $4/N$. The relationship between $\lambda_2$ and $h(H)$ can be seen through the following bounds: $$\label{eq:cheegerbound}
  \frac{\lambda_2}{2} \le h(H) \le \sqrt{\lambda_2{\left(2 \Delta - \lambda_2\right)}}.$$ Many other graph properties may be derived from the Laplacian spectrum as well (see, e.g., Refs. [@Newman2000; @Mohar1991]).

For a large network, finding the eigenvalues can be numerically expensive. However, hierarchies have a special structure which can be exploited for the evaluation of graph spectra. Here, we show 