Distance cube polynomials of Fibonacci and Lucas-run graphs

Abstract: The Fibonacci-run graphs $\mathcal{R}n$ are a family of an induced subgraph of hypercubes introduced by E\u{g}ecio\u{g}lu and Ir\v{s}i\v{c} in 2021. A cyclic version of $\mathcal{R}_n$, the Lucas-run graph $\mathcal{R}_n^l$, was also recently proposed (Jianxin Wei, 2024). We prove that the generating function previously given for the polynomial $D{\mathcal{R}_n}(x,q)$ which counts the number of hypercubes at a given distance in $\mathcal{R}_n$ was erroneous and determine its correct expression. We also consider Lucas-run graphs and prove the conjecture proposed by Jianxin Wei establishing the link between cube polynomials of $\mathcal{R}_n^l$ and $\mathcal{R}_n$.