A graph theoretic encoding of Lucas sequences

TL;DR

This paper introduces new graph classes that encode Lucas sequences and Dickson polynomials, providing combinatorial interpretations and generalizing known results about independent sets in paths and cycles.

Contribution

It presents a novel graph-theoretic framework that encodes Lucas sequences and Dickson polynomials, extending classical combinatorial results.

Findings

New graph classes encode Lucas sequences of both kinds.

Provides combinatorial interpretations of Dickson polynomial terms.

Generalizes known independent set counts in paths and cycles.

Abstract

Some well-known results of Prodinger and Tichy are that the number of independent sets in the $n$-vertex path graph is $F_{n+2}$, and that the number of independent sets in the $n$-vertex cycle graph is $L_n$. We generalize these results by introducing new classes of graphs whose independent set structures encode the Lucas sequences of both the first and second kind. We then use this class of graphs to provide new combinatorial interpretations of the terms of Dickson polynomials of the first and second kind.