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 Eğecioğlu and Iršič 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 ${\mathcal{R}}_n$.