Exercise

Exercise     For a strongly regular graph $\Gamma$ and a vertex $x$ of $\Gamma$, let $\Delta = \Gamma_2 ( x )$ be the subgraph consisting of the vertices at distance two from $x$. If $\Gamma$ has no triangles and spectrum $k^1 r^f s^g$, then show that $\Delta$ has spectrum $(k - \mu )^1 r^{f-k} s^{g-k} (- \mu )^{k-1}$. Conclude that $f \ge k$ and $g \ge k$, and that if equality holds then $\Delta$ is itself strongly regular. Determine all strongly regular graphs with $\lambda = 0$ and $f = k$.