A short proof of a theorem of Bang and Koolen

Let a graph Γ be locally a disjoint union of three copies of complete graphs K q − 1 and let Γ be cospectral with the Hamming graph H ( 3 , q ) . Bang and Koolen [S. Bang, J.H. Koolen, Graphs cospectral with H ( 3 , q ) which are disjoint unions of at most three complete graphs, Asian–Eur. J. Math. 1 (2008) 147–156] proved that if q > 3 , then Γ is isomorphic to H ( 3 , q ) . We present a short proof of this result.