## Characteristic polynomial of a regular graph

Let $\phi(x)$ be the characteristic polynomial of a $k$ -regular graph $X$ on $n$ vertices.
Let the graph $Y$
be obtained from $X$ by adding a new vertex and joining it to all vertices of $X$.

Prove that the characteristic
polynomial of $Y$ is $\frac{(x^{2}-kx-n)\phi(x)}{x-k}" alt="\frac{(x^{2}-kx-n)\phi(x)}{x-k}" />