Distance cube polynomials of Fibonacci and Lucas-run graphs

TL;DR

This paper corrects the generating function for distance cube polynomials of Fibonacci-run graphs and establishes a link between these polynomials for Fibonacci and Lucas-run graphs, advancing understanding of their combinatorial structure.

Contribution

It provides the correct expression for the distance cube polynomial of Fibonacci-run graphs and proves a conjecture relating these polynomials to Lucas-run graphs.

Findings

Corrected the generating function for $D_{\mathcal{R}_n}(x,q)$

Established a link between cube polynomials of Fibonacci and Lucas-run graphs

Proved a conjecture connecting the polynomials of the two graph families

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 $\mathcal{R}_n$.