Strongly Regular Square-free Graphs with μ=2

By a square in an undirected graph Γ, we mean a cyclex, y, z, wsuch thatxis not adjacent tozandyis not adjacent tow.Suppose that Γ is a strongly regular graph with μ=2, and assume that Γ does not contain a square. Pick any vertexxof Γ and let Γ′ denote the induced subgraph on the first subsconstituent of Γ with respect tox.It is known that Γ′ is strongly regular. We express the parameters of Γ and Γ′ in terms of two integer variables,pandh,which do not depend onx.We present feasibility conditions which must be satisfied bypandh,and show that, ifpsatisfies these, thenp=(f2i+1)2for some integeri(0≤i<∞), wheref0,f1,f2,... is the Fibonacci sequence, defined byf0=0,f1=1 andfj=fj-1+fj-2(2≤j≤∞).