subgraphs of $G$ - a problem that can be solved using Kocay’s Lemma in case of reconstruction from the vertex deck. Motivated by Kocay’s Lemma, we ask the following question. Suppose the coefficients $c_{i_1}(G), c_{i_1}(G), \ldots , c_{i_k}(G)$ are known, and $i_1 + i_2 + \ldots + i_k \geq v(G)$. If the coefficients $c_{i_j}; 1 \leq j \leq k$ are multiplied, can we get some information about the spanning subgraphs of $G$? This is especially tempting if $i_1+i_2 + \ldots + i_k = v(G)$, since the product is expected to have some relationship with the disconnected spanning elementary subgraphs of $G$. This idea is explored in Section \[sec-poly\].

In Section \[sec-whitney\], we present a very simple new proof of Whitney’s subgraph expansion theorem, again based on Kocay’s lemma. We then present a more direct argument to compute the characteristic polynomial of a graph from its vertex deck, based on our formulation of Whitney’s theorem.

Ulam’s conjecture and the $\mathcal{N}$-matrix {#sec-n-rec1}
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Let $\Lambda(G) = \{\Lambda_i; \,i \in [1,I]\}$ be the set of distinct isomorphism types of nonempty induced subgraphs of $G$. We call this the $\Lambda$-deck of $G$. Let $\mathcal{N}(G)=(N_{ij})$ be an $I$ x $I$ incidence matrix where $N_{ij}$ is 