Strongly regular square-free graphs with \(\mu=2\) (Q1357655)

A square-free graph \(\Gamma\) contains no cycle \(x,y,z,w\) such that \(x\) is not adjacent to \(z\) and \(y\) is not adjacent to \(w\). The paper investigates the possible orders of square-free strongly regular graphs with \(\mu=2\). The main result is a long series of strong number-theoretical conditions on the number of vertices of such graphs. Based on these conditions, the author was able to run a computer search which showed that any square-free strongly regular graph with \(\mu=2\) has at least \(5.8 \times 10^{58}\) vertices.