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
$$
\operatorname{FWI}^*(\mathscr{T})=\sum_{n,m\in E(\mathscr{T})}[\deg F_n^2-\deg F_m^2],
$$
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 ofFibonacci words, providing new insights into their algebraic and topological characteristics. Additionally, we present analytical expressions involving Fibonacci numbers and their generating functions, supported by Binet’s formula, to facilitate computation of these indices. The theoretical developments are illustrated with examples, including detailed constructions of Fibonacci word trees and their degree distributions. This work opens avenues for further investigation of word-based graph invariants and their applications in combinatorics and theoretical computer science.
Keywords:Fibonacci, word, trees, topological, indices, irregularity.
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