On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds

TL;DR

This paper investigates bounds on topological indices in trees, specifically the Albertson and Sigma indices, deriving formulas and bounds related to Fibonacci degree sequences and connecting these indices to other graph invariants.

Contribution

It provides a precise formula for the Albertson index in trees with Fibonacci degree sequences and establishes new bounds and relationships with other topological indices.

Findings

Derived a formula for the Albertson index in Fibonacci degree trees

Established bounds for minimum and maximum Albertson indices

Connected the Albertson index to Zagreb and forgotten indices

Abstract

In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $\sigma$. The Sigma index was defined through the following relationship: \[ \sigma(G)=\sum_{uv\in E(G)}\left( d_u(G)-d_v(G) \right)^2. \] We establish a precise formula for the Albertson index of a tree $T$ of order $n$ with a Fibonacci degree sequence $\mathscr{D} = (F_3, \dots, F_n)$. Additionally, we derive bounds for the minimum and maximum Albertson indices ($\irr_{\min}$ and $\irr_{\max}$) across various tree structures. Propositions and lemmas provide upper and lower bounds, incorporating parameters such as the maximum degree $ \Delta$, minimum degree $\delta$. We further relate the Albertson index to the second Zagreb index $M_2(T)$ and the forgotten index $F(T)$, establishing a new upper bound.