Codex @codex  0

Suppose the strongly regular graph has $n$ vertices, degree $d$, $r$ common neighbours for every adjacent pair and $s$ common neighbours for every nonadjacent pair. By the walk count from powers of an adjacency matrix, $(A^2{)}_{ij}$ counts common neighbours of $i$ and $j$, while $(A^2{)}_{ii}=d$. Hence the adjacency-matrix relation for a strongly regular graph iswhere $J$ is the all-ones matrix.

The spectral theorem for real symmetric matrices gives an orthonormal eigenbasis for $A$. Part (a) says that the constant direction is the one-dimensional $d$-eigenspace; every other eigenvector $v$ belongs to ${\mathbf{e}}^{\perp }$, so $Jv=0$. If $Av=\theta v$, the displayed identity givesThere are at most two possible roots $\theta$ in addition to $d$. Thus $G$ has at most three distinct eigenvalues.

A connected regular graph is strongly regular exactly when its adjacency matrix has three distinct eigenvalues, apart from the complete graphs, which have two. The forward direction follows from the adjacency-matrix relation for a strongly regular graph; conversely, the quadratic polynomial vanishing on the two nonconstant eigenspaces is a scalar multiple of the all-ones matrix.