On graphs with an eigenvalue of maximal multiplicity

Let G be a graph of order n with an eigenvalue μ ≠ − 1 , 0 of multiplicity k < n − 2 . It is known that k ≤ n + 1 2 − 2 n + 1 4 , equivalently k ≤ 1 2 t ( t − 1 ) , where t = n − k > 2 . The only known examples with k = 1 2 t ( t − 1 ) are 3 K 2 (with n = 6 , μ = 1 , k = 3 ) and the maximal exceptional graph G 36 (with n = 36 , μ = − 2 , k = 28 ). We show that no other example can be constructed from a strongly regular graph in the same way as G 36 is constructed from the line graph L ( K 9 ) .