graphs. Its main advantage comes from the convenience of working with the algebra at the level of adjacency matrices and Laplacians, which in turn makes the computation of important properties of such graphs straightforward.

We now present some properties of the hierarchical product which make it an attractive formalism for practical applications in quantum networking.

### Structural Properties {#sec:struc}

At the level of adjacency matrices, the hierarchical product is *associative*. Let $A, B, C$ be three adjacency matrices. Then, $${\left(A{\mathbin{\Pi}}B\right)}{\mathbin{\Pi}}C = A{\mathbin{\Pi}}{\left(B{\mathbin{\Pi}}C\right)}.$$ For a proof, we refer the reader to Ref. [@Barriere2009].

Associativity implies that a product of multiple graphs does not depend on the order of evaluation. Therefore, we can unambiguously take the hierarchical product over many graphs to produce a graph of the form $G_k{\mathbin{\Pi}}G_{k-1}{\mathbin{\Pi}}\cdots{\mathbin{\Pi}}G_1$. We will refer to such graphs as *hierarchies*, and the $i$-th graph in the product $G_i$ as the $i$-th level of the hierarchy, enumerated from the bottom level upwards (symbolically, from right to left). In particular, if all $G_i$ are equal to some graph $G$, then we write $$\begin{aligned}
\label{eq:unwHPcomp}
G^{{\mathbin{\Pi}}k} := \underbrace{G {\mathbin{\Pi}}\cdots G\, {\mathbin{\Pi}}}_{k-1\mathrm{\ times}} G.\end{aligned}$$ and refer to $G^{{\mathbin{\Pi}}k}$ as a depth-$k$ (or $k$-level) hierarchy.

Note that the hierarchical product 