On a family of diamond-free strongly regular graphs

The existence of a partial quadrangle ${\mathsf{PQ}}(s, t, \mu)$ is equivalent to the existence of a diamond-free strongly regular graph ${\mathsf{SRG}}(1+s(t+1)+s^2t(t+1)/\mu, s(t+1), s-1, \mu)$. Recently, it is shown that there exists a ${\mathsf{PQ}}(2, (n^3+3n^2-2)/2, n^2+n)$ if and only if $n\in\{1, 2, 4\}$. Let $\mathcal{S}$ be a ${\mathsf{PQ}}(3,(n+3)(n^2-1)/3, n^2+n)$ such that for every two non-collinear points $p_1$ 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 $n=10$.