Horadam cubes

TL;DR

This paper introduces a new family of graphs called Horadam cubes, which generalize Fibonacci and metallic cubes using a second-order linear recurrence, and explores their structural, metric, and enumerative properties.

Contribution

It defines Horadam cubes based on a second-order recurrence and analyzes their properties, extending known results from Fibonacci and metallic cubes.

Findings

Horadam cubes preserve key properties of Fibonacci and metallic cubes

Recursive decomposition and grid decomposition are established

Properties like edges, degrees, and Hamiltonian paths are analyzed

Abstract

We define and investigate a new three-parameter family of graphs that further generalizes the Fibonacci and metallic cubes. Namely, the number of vertices in this family of graphs satisfies Horadam recurrence, a linear recurrence of second order with constant coefficients. It is shown that the new family preserves many appealing and useful properties of the Fibonacci and metallic cubes. In particular, we present recursive decomposition and decomposition into grids. Furthermore, we explore metric and enumerative properties such as the number of edges, distribution of degrees, and cube polynomials. We also investigate the existence of Hamiltonian paths and cycles.