Let$\displaystyle \phi(x)$be the characteristic polynomial of a $\displaystyle k$ -regular graph $\displaystyle X$ on $\displaystyle n$ vertices.

Let the graph $\displaystyle Y$ be obtained from $\displaystyle X$ by adding a new vertex and joining it to all vertices of $\displaystyle X$.

Prove that the characteristic polynomial of $\displaystyle Y$ is $\displaystyle \frac{(x^{2}-kx-n)\phi(x)}{x-k}$