on finite groups all of whose bi-cayley graphs of bounded valency are integral

Let k ≥ 1 be an integer and ik 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 1 ≤ |s| ≤ k is integral. let k ≥ 3. we prove that a finite group g belongs to ik if and only if g ∼= z3, z r 2 for some integer r, or s3.