suppose that $G$ is strongly regular graph $srg(n,k,\lambda ,\mu)$,suppose $r$ is the second largest eigenvalue of adjacency matrix,prove that $r< \frac{k}{2}$ ,in addition $r\leq \frac{1}{2}(k-1)$ unless $G=C_5$ .
we know that if $G$ is not complete then $r=\frac{1}{2}((\lambda-\mu)+\sqrt{(\lambda-\mu)^{2}+4(k-\mu)})$
I supposed that $2r<k$ with some calculation and using $k\geq 2\lambda -\mu +3$ I arrived $k\mu +k>k$ now if this inequality is right then I will back this way,but I couldn't show that this inequality is right,and maybe it is not right at all,
it will be great to help me with ,thanks.