of a graph from the characteristic polynomials of all induced proper subgraphs. Finally, we present a new short proof of Whitney’s subgraph expansion theorem, and demonstrate the reconstructibility of the characteristic polynomial of a graph using Whitney’s theorem.

Notation
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We consider only finite simple graphs in this paper. Let $G$ be a graph with vertex set $VG$ and edge set $EG$. The number of vertices of $G$ is denoted by $v(G)$ and the number of edges is denoted by $e(G)$. When $VG=\emptyset $, we denote $G$ by $\Phi$, and call the graph a [*null graph*]{}. When $EG=\emptyset $, we call the graph an [*empty graph*]{}. When $F$ is a subgraph of $G$, we write $F\subseteq G$, and when $F$ is a proper subgraph of $G$, we write $F\subsetneq G$. The subgraph of $G$ induced by $S \subseteq VG$ is the subgraph whose vertex set is $S$ and whose edge set contains all the edges having both end vertices in $S$. It is denoted by $G_S$. The subgraph of $G$ induced by $VG - S$ is denoted by $G-S$, or simply $G-u$ if $S=\{u\}$. A subgraph of $G$ with vertex set $V\subseteq VG$ and edge set $E\subseteq EG$ is denoted 