On a family of diamond-free strongly regular graphs

The existence of a partial quadrangle $\mathsf{P}\mathsf{Q}(s,t,\mu )$ is equivalent to the existence of a diamond-free strongly regular graph $\mathsf{S}\mathsf{R}\mathsf{G}(1+s(t+1)+s^{2}t(t+1)\mathrm{/}\mu ,s(t+1),s-1,\mu )$. Recently, it is shown that there exists a $\mathsf{P}\mathsf{Q}(2,(n^3+3n^2-2)\mathrm{/}2,n^2+n)$ if and only if $n\in \{1,2,4\}$. Let $\mathcal{S}$ be a ${$\mathsf{P}\mathsf{Q}$}(3,(n+3)(n^2-1)/3, n^2+n)$suchthatforeverytwonon-collinearpoints$p_1$and$p_2$, there is a point $q$ non-collinear with $p_1$, $p_2$, and all points collinear with both $p_1$ and $p_2$. In this article, we establish that $\mathcal{S}$ exists only for $n\in \{-2,2,3\}$ and probably $n=10$.