On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s   satisfies ns⩽2ns⩽2 and ns*s≠2ns*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//〈t〉G//〈t〉, where G is a finite group and t an involution of G   weakly closed in CG(t)CG(t).Groups G with an involution t   weakly closed in CG(t)CG(t) have been described in Glauberman's Z*Z*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.