On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds
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| Format: | Preprint |
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2025
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| _version_ | 1866916792502321152 |
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| author | Hamoud, Jasem Belov-Kanel, Alexei Abdullah, Duaa |
| author_facet | Hamoud, Jasem Belov-Kanel, Alexei Abdullah, Duaa |
| contents | In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $σ$. The Sigma index was defined through the following relationship: \[ σ(G)=\sum_{uv\in E(G)}\left( d_u(G)-d_v(G) \right)^2. \]
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 $ Δ$, minimum degree $δ$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds Hamoud, Jasem Belov-Kanel, Alexei Abdullah, Duaa Combinatorics 05C05, 05C12, 05C35, 68R10 G.2.2 In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $σ$. The Sigma index was defined through the following relationship: \[ σ(G)=\sum_{uv\in E(G)}\left( d_u(G)-d_v(G) \right)^2. \] 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 $ Δ$, minimum degree $δ$. 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. |
| title | On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds |
| topic | Combinatorics 05C05, 05C12, 05C35, 68R10 G.2.2 |
| url | https://arxiv.org/abs/2506.11223 |