The Fibonacci cube Γ n is the subgraph of the hypercube Q n induced by vertices with no consecutive 1s. Recently Jianxin Wei and Yujun Yang introduced a one parameter generalization, Fibonacci p -cubes Γ n p , which are subgraphs of hypercubes induced by strings where there is at least p consecutive 0s between two 1s. In this paper we first prove the expression conjectured by the authors for the cube polynomial of Γ n p . By a totally different method we then determine a generalization, the distance cube polynomial. We also complete the invariants investigated in the original paper by three new ones, the Mostar index Mo ( Γ n p ) , the Wiener index W ( Γ n p ) and the Irregularity irr ( Γ n p ) .

When [Formula: see text] Ramras [J. Combin. Theory Ser. B, 54 (1992), pp. 239–248] proved, by a counting argument, that for any isometrically embedded tree [Formula: see text] on [Formula: see text] edges in [Formula: see text] there exists a group of translations [Formula: see text] such that [Formula: see text] is a vertex partition of [Formula: see text]. Considering a more general context we are able to give an explicit construction of [Formula: see text] and can construct nongroup vertex partitions by isometric trees. We also extend this problem to vertex partition of [Formula: see text] by translates of an isometrically embedded tree on [Formula: see text] edges for any [Formula: see text].

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of G is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023–1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if n is even and C a connected component of $P_m\times P_n$, where m is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.