We set $N = n^2$ and
$$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Verify that, in the case where $n = 3$, $J_N^{(2)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge (the edges count whether they are dashed or solid), and equals 0 otherwise.