Let $G$ be a graph of maximum degree $\Delta$. Show the following:

(i) Every eigenvalue $\lambda$ of $G$ satisfies $|\lambda| \leqslant \Delta$.

(ii) If $G$ is regular then $\Delta$ is an eigenvalue.

(iii) If $G$ is regular and connected then the multiplicity of $\Delta$ as an eigenvalue is 1 .

(iv) If $G$ is regular and not connected then the multiplicity of $\Delta$ as an eigenvalue is greater than 1 .

Let $A$ be the adjacency matrix of the Petersen graph. Explain why $A^{2}+A-2 I=J$, where $I$ is the identity matrix and $J$ is the all-1 matrix. Find, with multiplicities, the eigenvalues of the Petersen graph.