Nullity of a undirected graph with a pendant edge

Question

Let $G$ be a graph with a vertex $v$ having degree 1. Let $G_1$ be the induced subgraph of $G$ after removing $v$, and $G_2$ be the induced graph of $G$ after removing $v$ along with its adjacent vertex. Let $\phi(G)$ denote the characteristic polynomial (corresponding to the adjacency matrix) of $G$. It is well known that $$\phi(G)=\lambda\phi(G_1)-\phi(G_2).$$ Using above relation how can we prove that the nullity of $G$ equals to the nullity of $G_2?$