Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs

TL;DR

This paper explores conjectured relationships between integer sequences and graph parameters in linear Jaco graphs, highlighting the role of Golden ratio-like functions, with an emphasis on experimental methods and the challenge of proving these conjectures.

Contribution

It introduces new conjectures linking integer sequences and graph parameters in linear Jaco graphs, emphasizing the role of Golden ratio-like functions and proposing an experimental approach.

Findings

Conjectured relations between integer sequences and graph parameters.

Golden ratio-like floor functions influence graph structural properties.

Experimental methodology suggests potential for future proofs.

Abstract

This experimental study presents some interesting conjectured relations between some integer sequences and certain graph parameters of the family of linear Jaco graphs $J_n(x)$ where $n = 1,2,3,\dots$. It appears that $\textit{Golden ratio}$-like floor function terms play an important role in the analysis of the graph structural properties of the family of linear Jaco graphs. The experimental methodology to obtain the conjectures is indeed trivial. However, it is the author's view that the proofs or disproofs of the conjectures may be challenging.