On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds

In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{i}\mathrm{r}\mathrm{r}$ and the Sigma index $\sigma$. The Sigma index was defined through the following relationship: $$\sigma (G)=\sum _{uv\in E(G)}{(d_u(G)-d_v(G))}^2\mathrm{.}$$ We establish a precise formula for the Albertson index of a tree $T$ of order $n$ with a Fibonacci degree sequence $\mathcal{D}=(F_3,\dots ,F_n)$. Additionally, we derive bounds for the minimum and maximum Albertson indices (${\text{irr}}_{\min }$ and ${\text{irr}}_{\max }$) across various tree structures. Propositions and lemmas provide upper and lower bounds, incorporating parameters such as the maximum degree $\mathrm{\Delta }$, 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.