On graphs with an eigenvalue of maximal multiplicity

Let GG be a graph of order nn with an eigenvalue μ≠−1,0μ≠−1,0 of multiplicity k2t=n−k>2. The only known examples with k=12t(t−1) are 3K23K2 (with n=6n=6, μ=1μ=1, k=3k=3) and the maximal exceptional graph G36G36 (with n=36n=36, μ=−2μ=−2, k=28k=28). We show that no other example can be constructed from a strongly regular graph in the same way as G36G36 is constructed from the line graph L(K9)L(K9).