Metallic cubes

TL;DR

This paper introduces a new family of graphs called metallic cubes, generalizing Fibonacci and Pell graphs, and explores their structural, enumerative, and metric properties, highlighting their potential applications.

Contribution

It defines and analyzes a new recursive graph family, establishing their subgraph relation to hypercubes and computing key invariants and properties.

Findings

All graphs are induced subgraphs of hypercubes

Computed metric invariants and degree distribution

Established Hamiltonicity and structural decompositions

Abstract

We study a recursively defined two-parameter family of graphs which generalize Fibonacci cubes and Pell graphs and determine their basic structural and enumerative properties. In particular, we show that all of them are induced subgraphs of hypercubes and present their canonical decompositions. Further, we compute their metric invariants and establish some Hamiltonicity properties. We show that the new family inherits many useful properties of Fibonacci cubes and hence could be interesting for potential applications. We also compute the degree distribution, opening thus the way for computing many degree-based topological invariants. Several possible directions of further research are discussed in the concluding section.