Let $G$ be a connected $d$-regular graph.

(a) Show that $d$ is an eigenvalue of $G$ with multiplicity 1 and eigenvector

$$e=(11 \ldots 1)^{T}$$

(b) Suppose that $G$ is strongly regular. Show that $G$ has at most three distinct eigenvalues.

(c) Conversely, suppose that $G$ has precisely three distinct eigenvalues $d, \lambda$ and $\mu$. Let $A$ be the adjacency matrix of $G$ and let

$$B=A^{2}-(\lambda+\mu) A+\lambda \mu I .$$

Show that if $v$ is an eigenvector of $G$ that is not a scalar multiple of $e$ then $B v=0$. Deduce that $B$ is a scalar multiple of the matrix $J$ whose entries are all equal to one. Hence show that, for $i \neq j,\left(A^{2}\right)_{i j}$ depends only on whether or not vertices $i$ and $j$ are adjacent, and so $G$ is strongly regular.

(d) Which connected $d$-regular graphs have precisely two eigenvalues? Justify your answer.