On a family of diamond-free strongly regular graphs

Abstract

The existence of a partial quadrangle $\mathsf{PQ}(s,t,\mu )$ is equivalent to the existence of a diamond-free strongly regular graph $\mathsf{SRG}(1+s(t+1)+s^{2}t(t+1)/\mu ,s(t+1),s-1,\mu )$. 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 $n=10$.