The size of $k$-th order generalized Fibonacci cubes

TL;DR

The paper studies the edge count $|E(Γ^{(k)}_{n})|$ of the $k$-th order generalized Fibonacci cubes $Γ^{(k)}_{n}$, graphs obtained from the hypercube $Q_{n}$ by forbidding any string with $k$ consecutive $1$s. Building on the $k$-th order Fibonacci numbers $F^{(k)}_{n}$, it derives two closed forms: a convolution form $|E(Γ^{(k)}_{n})| = \sum_{j=1}^{k-1} \left( j \sum_{i=k-1}^{n+k-1-j} F^{(k)}_{i} F^{(k)}_{n+2k-2-j-i} \right)$ and a linear form $|E(Γ^{(k)}_{n})| = \frac{1}{2(2k)^k - (k+1)^{k+1}} \sum_{j=0}^{k-1} (n A_j + B_j) F^{(k)}_{n+j}$, with the coefficients $A_j,B_j$ specified in the paper. These results generalize known formulas for $k=2$ and $k=3$ and rely on a blend of combinatorial decompositions and properties of the $k$-th order Fibonacci numbers. The work also discusses extensions to Fibonacci $p$-cubes and outlines future directions for convolution and linear formulas in broader Fibonacci-like cube families.

Abstract

Let $k\geq2$. Then the $k$-th order Fibonacci cube $Γ^{(k)}_{n}$ is the subgraph of the hypercube $Q_{n}$ induced by vertices without $k$ consecutive $1$s. The case $k=2$ corresponds to the classic Fibonacci cube $Γ_{n}$. There are three kinds of calculation formulas of the size of $Γ_{n}$: the iteration form $|E(Γ_{n})|=|E(Γ_{n-1})|+|E(Γ_{n-2})|+F_{n}$ (Hsu, 1993), %iteration form the convolution form $|E(Γ_{n})|=\mathop{\sum}\limits_{i=1}^{n}F_{i}F_{n-i+1}$ (Klavžar, 2005) %convolution form and the linear form $|E(Γ_{n})|=\frac{nF_{n+1}+2(n+1)F_{n}}{5}$ (Munarini et al., 2001). %linear form Belbachir and Ould-Mohamed (2020) studied the iteration and convolution formulas of the size of $Γ^{(3)}_{n}$. Very recently, Mollard (2025) deduced the iteration formula of the size of $Γ^{(k)}_{n}$ for $k\geq2$. In this paper, we give the the formulas of convolution and linear forms of $|E(Γ^{(k)}_{n})|$ for all $k\geq2$. Specifically, we obtain the formula of $|E(Γ^{(k)}_{n})|$ in terms of convolved $k$-th order Fibonacci numbers and the formula of $|E(Γ^{(k)}_{n})|$ of linear expression of $k$ consecutive $k$-th order Fibonacci numbers.