On strongly regular graphs with parameters (k,0,2) and their antipodal double covers

Abstract

Let $\Gamma$ be a strongly regular graph with parameters $(k, \lambda, \mu)=(q^{2}+1,0,2)$ admitting $G(\cong PGL(2, q)2)$ as one point stabilizer for odd prime power $q$. We show that if $G$ stabilizes a vertex $x$ of $\Gamma$ and acts on $\Gamma_{2}(x)$ transitively, then $q=3$ holds and $\Gamma$ is the Gewirtz graph. Moreover it is shown that an antipodal double cover whose diameter 4 of a strongly regular graph with parameters $(k, 0, 2)$ is reconstructed from a symmetric association scheme of class 6 with parameters $p_{j.k}^{i}(0\leq i, j, k\leq 6)$ in the Section 3.