In a percolative problem, I have noticed that all of the nodes of my system are connected to 3 other nodes. I started drawing a bit and realized that this could look like a honeycomb lattice.
The adjacency matrix is built by assigning the labels $i\in{0,...,n}$ to each node, with $n$ the number of nodes, and assigning $A_{ij} = 1$ if $i$ and $j$ are connected and 0 otherwise.
In my case the adjacency matrix has 3 1's on every line (and is symmetric by definition).
My question is : What properties should the adjacency matrix possess to truly represent the topology of a honeycomb lattice ? Is the fact that the sum of all its columns/lines = 3 enough ? Should I be able to tell something about the spectrum of its eigenvalues ?
All input appreciated
It does not seem enough as shown by the following graph: $$\left(\begin{matrix} 0&1&1&1&0&0&0&0&0&0 \\ 1&0&1&1&0&0&0&0&0&0 \\ 1&1&0&0&1&0&0&0&0&0 \\ 1&1&0&0&1&0&0&0&0&0 \\ 0&0&1&1&0&1&0&0&0&0 \\ 0&0&0&0&1&0&1&0&0&1 \\ 0&0&0&0&0&1&0&1&1&0 \\ 0&0&0&0&0&0&1&0&1&1 \\ 0&0&0&0&0&0&1&1&0&1 \\ 0&0&0&0&0&1&0&1&1&0 \\ \end{matrix}\right) $$