the characteristic polynomial $P(G;\lambda)$ cannot always be computed from $\mathcal{N}(G)$. For example, the only non-quasi-separable induced subgraphs of any tree $T$ are $K_1$ and $K_2$, so $P(T;\lambda)$ cannot be computed from $\mathcal{N}(T)$. Therefore, we omit the restriction of non-quasi-separability on the induced subgraphs used in the construction of the incidence matrix. We then investigate which invariants of a graph are determined by its $\mathcal{N}$-matrix.

The Sections  \[sec-n-rec1\] and  \[sec-nmatrix\] are devoted to the study of reconstruction from the $\mathcal{N}$-matrix. In Section \[sec-n-rec1\], we formally define the $\mathcal{N}$-matrix, and the related concept of the edge poset of induced subgraphs of a graph. We then prove several basic results on the relationship between the $\mathcal{N}$-matrix, the edge labelled poset and reconstruction. In particular, we show that Ulam’s conjecture is true if and only if the $\mathcal{N}$-matrix itself is a complete graph invariant. We then prove that Ulam’s conjecture is true if and only if the edge labelled poset has no non-trivial automorphisms. We also prove the $\mathcal{N}$-matrix reconstructibility of trees and forests.

In Section \[sec-nmatrix\] we compute several invariants of a graph from its $\mathcal{N}$-matrix. We prove that the characteristic polynomial $P(G;\lambda)$ of a graph $G$, its rank polynomial $R(G;x,y)$, the number 