(in Theorem \[th:specMain\]) that if the spectra of the base graphs $L_i$ are known, then one can derive the spectrum of the $k$-level hierarchy efficiently using a recursive procedure. We first present two lemmas. The first lemma generalizes Theorem 3.10 from Ref. [@Barriere2009], which states that the characteristic polynomial $\phi_P(x)$ ${\left(= \det{\left[x{\mathbbm{1}}- P\right]}\right)}$ of an unweighted hierarchical product of adjacency matrices $A$, $B$ is given by $$\label{eq:thm310}
  \phi_P(x) = \phi_{B{^\prime}}{\left(x\right)}^{n_A}\phi_A{\left(\frac{\phi_{B}{\left(x\right)}}{\phi_{B{^\prime}}{\left(x\right)}}\right)},$$ where $A{^\prime}$ (resp. $B{^\prime}$) is the matrix $A$ (resp. $B$) with the first row and first column removed, and $n_A = |G_A|$ is the order of the graph $A$. In fact, \[eq:thm310\] applies to Laplacians as well as adjacency matrices. The lemma below further generalizes this statement to a weighted product of Laplacians.

\[lm:spec1\] Let $K$ and $L$ be two graph Laplacians with characteristic polynomials given by $\phi_{K}(x)$ and $\phi_L(x)$, respectively. Then, the characteristic polynomial $\phi_{{\mathbin{\Pi}}}(x)$ of the hierarchical product $K {\mathbin{\Pi}}_\alpha L$ is given by $$\label{eq:HPchar}
  \phi_{{\mathbin{\Pi}}} {\left(x\right)} = {\left[\alpha \phi_{L{^\prime}}{\left(x\right)}\right]}^{n_K}\phi_K{\left(\frac{1}{\alpha}\frac{\phi_{L}{\left(x\right)}}{\phi_{L{^\prime}}{\left(x\right)}}\right)},$$ where $n_k = \dim {\left\{K\right\}}$, and $L{^\prime}$ is defined similar to $A{^\prime}$ and $B{^\prime}$ above.

Denote the spectra of $K$ and $L$ by ${\left\{\kappa_j\right\}}$ and ${\left\{\lambda_j\right\}}$, respectively. Recall that the $\alpha$-weighted hierarchical product may be written as $$K{\mathbin{\Pi_\alpha}}L 