On finite groups all of whose bi-Cayley graphs of bounded valency are integral

Let k\geq ۱ be an integer and \mathcal{I}_k be‎ ‎the set of all finite groups G such that every bi-Cayley graph \BCay(G,S) of G with respect to‎ ‎subset S of length ۱\leq |S|\leq k is integral‎. ‎Let k\geq ۳‎. ‎We prove that a finite group G belongs to \mathcal{I}_k if and‎ ‎only if G\cong\Bbb Z_۳‎, ‎\Bbb Z_۲^r for some integer r‎, ‎or S_۳‎.

Keywords:

‎bi-Cayley graph‎ , ‎Integer eigenvalues‎ , ‎Irreducible representation