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}" />