Investigating independent subsets of graphs, with Mathematica

TL;DR

This paper demonstrates how Mathematica can be used to explore combinatorial properties of independent subsets in graphs, revealing connections with Fibonacci and Lucas cubes through enumeration and structural analysis.

Contribution

It introduces a computational approach using Mathematica to investigate independent subsets of graph powers and their relation to well-known combinatorial structures.

Findings

Enumeration of independent subsets of powers of paths and cycles.

Identification of correspondences with Fibonacci and Lucas Cubes.

Structural analysis of partially ordered sets of independent subsets.

Abstract

With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to highlight the correspondence with other combinatorial objects with the same cardinality. Then we will study the structures obtained by ordering properly independent subsets of paths and cycles. We will approach some enumeration problems on the resulting partially ordered sets, putting in evidence the correspondences with structures known as Fibonacci and Lucas Cubes.