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

This paper introduces the concept of the Fibonacci Word Index $\mathrm{FWI}$, 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 $\mathrm{FWI}$ and its variants, including the total irregularity and modified Fibonacci Word Index where it defined as $${\mathrm{FWI}}^{\ast }(T)=\sum _{n,m\in E(\mathcal{T})}[deg{F}_n^2-deg{F}_m^2],$$ 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.