Projection vectors of Fibonacci cubes

Definition(s): The Cartesian product G2H of graphs G and H is the graph with the vertex set V (G) × V (H) where a vertex (a, x) is adjacent to a vertex (b, y) whenever ab ∈ E(G) and x = y, or a = b and xy ∈ E(H). Hamming graphs are Cartesian products of complete graphs. Let H be a subgraph of the Cartesian product G = ∏k i=1Gi, k ≥ 2. Let e = hh ′ be an edge of H, where h = (h1, h2,..., hk) and h ′ = (h′1, h′2,..., h′k). Then there exists exactly one i such that hih′i ∈ E(Gi), while hj = h′j for j 6 = i. We will say that the edge e is of type i. For i = 1, 2,..., k let Ei(H;G) = {e ∈ E(H) | e is of type i}, set ei(H;G) = |Ei(H;G)|, and let v(H;G) = (e1(H;G), e2(H;G),..., ek(H;G)) be the projection vector of the subgraph H of the Cartesian product G. A Fibonacci string of length n is a binary string a1a2... an with aiai+1 = 0, 1 ≤ i < n. In other words, a Fibonacci string is a binary string of length n with no two consecutive ones. The Fibonacci cube Γn, n ≥ 1, has all Fibonacci strings of length n as vertices, two vertices being adjacent whenever they differ in exactly one coordinate. In [1] the following result was proved for the Fibonacci sequence (Fn) n=1