Irregularity and Topological Indices in Fibonacci Word Trees and Modified Fibonacci Word Index

TL;DR

This paper introduces the Fibonacci Word Index, a new topological measure for Fibonacci word trees, analyzing their irregularity and establishing inequalities relating these indices to tree degrees.

Contribution

It defines the Fibonacci Word Index and its variants, extending graph invariants to Fibonacci word trees and exploring their structural properties.

Findings

Defined the Fibonacci Word Index and its variants.

Established inequalities relating indices to maximum degree.

Extended graph invariants to Fibonacci word trees.

Abstract

This paper introduces the concept of the Fibonacci Word Index $\operatorname{FWI}$, a novel topological index derived from the Albertson index, applied to trees constructed from Fibonacci words. Building upon the classical Fibonacci sequence and its generalizations, we explore the structural properties of Fibonacci word trees and their degree-based irregularity measures. We define the $\operatorname{FWI}$ and its variants, including the total irregularity and modified Fibonacci Word Index where it defined as \[ \operatorname{FWI}^*(T)=\sum_{n,m\in E(\mathscr{T})}[deg F_n^2-deg F_m^2], \] and establish foundational inequalities relating these indices to the maximum degree of the underlying trees. Our results extend known graph invariants to the combinatorial setting of Fibonacci words, providing new insights into their algebraic and topological characteristics.