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 $\displaystyle \frac{(x^{2}-kx-n)\phi(x)}{x-k}$