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A strongly regular graph with parameters $(k,a,b)$ is a simple graph in which each vertex has degree $k$, adjacent vertices have exactly $a$ common neighbours, and distinct nonadjacent vertices have exactly $b$ common neighbours. Its order $n$ is often included separately as a fourth parameter.

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.