the number of induced subgraphs of $\Lambda_i$ that are isomorphic to $\Lambda_j$. Thus $N_{ii}$ is 1 for all $i \in [1,I]$. We call an invariant of a graph [*$\mathcal{N}$-matrix reconstructible*]{} if it can be computed from the (unlabelled) $\mathcal{N}$-matrix of the graph.

As an example, the ladder graph $L_3$ and its collection of distinct induced subgraphs with nonempty edge sets are shown in Figure 1. Below each graph (except $L_3$) is shown its multiplicity as an induced subgraph in $L_3$ and its name.

\[fig-l3\]

(200,400) (0,100) (200,100) (0,400) (200,400) (100,200) (100,300) (0,100)[(1,0)[200]{}]{} (0,100)[(0,1)[300]{}]{} (0,400)[(1,0)[200]{}]{} (200,100)[(0,1)[300]{}]{} (0,100)[(1,1)[100]{}]{} (200,100)[(-1,1)[100]{}]{} (0,400)[(1,-1)[100]{}]{} (200,400)[(-1,-1)[100]{}]{} (100,200)[(0,1)[100]{}]{} (0,0)[$L_3 = \Lambda_9$]{}

(100,400) (100,150) (100,300) (100,150)[(0,1)[150]{}]{} (50,0)[9$\Lambda_1$]{}

(200,400) (0,150) (200,150) (100,300) (0,150)[(1,0)[200]{}]{} (50,0)[6$\Lambda_2$]{}

(200,400) (0,150) (150,150) (0,300) (0,150)[(1,0)[150]{}]{} (0,150)[(0,1)[150]{}]{} (10,0)[12$\Lambda_3$]{}

(200,400) (0,150) (200,150) (100,300) (0,150)[(1,0)[200]{}]{} (0,150)[(2,3)[100]{}]{} (200,150)[(-2,3)[100]{}]{} (50,0)[2$\Lambda_4$]{}

(50,400) (50,100) (50,200) (50,300) (50,400) (50,100)[(0,1)[100]{}]{} (50,200)[(0,1)[100]{}]{} (50,300)[(0,1)[100]{}]{} (0,0)[6$\Lambda_5$]{}

(200,400) (0,150) (200,150) (100,250) (100,350) (0,150)[(1,0)[200]{}]{} (0,150)[(1,1)[100]{}]{} (200,150)[(-1,1)[100]{}]{} (100,250)[(0,1)[100]{}]{} (50,0)[6$\Lambda_6$]{}

(200,400) (0,150) (200,150) (0,350) (200,350) (0,150)[(1,0)[200]{}]{} (0,150)[(0,1)[200]{}]{} (0,350)[(1,0)[200]{}]{} (200,150)[(0,1)[200]{}]{} (50,0)[3$\Lambda_7$]{}

(200,400) (0,150) (200,150) (0,350) (200,350) (300,250) (0,150)[(1,0)[200]{}]{} (0,150)[(0,1)[200]{}]{} (0,350)[(1,0)[200]{}]{} (200,150)[(0,1)[200]{}]{} (200,150)[(1,1)[100]{}]{} (200,350)[(1,-1)[100]{}]{} (50,0)[6$\Lambda_8$]{}

The rows and the columns of $\mathcal{N}$-matrix of $L_3$ are both indexed by $\Lambda_1$ to $\Lambda_9$. The $\mathcal{N}$-matrix of $L_3$ is shown below. $$\mathcal{N}(L_3) = 
\begin{pmatrix}
1 & 0 