Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.03364 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915832814108672 |
|---|---|
| author | Hamoud, Jasem |
| author_facet | Hamoud, Jasem |
| contents | The Sombor index, a degree-based topological descriptor introduced by Gutman in 2021, lacks closed-form expressions for complex hierarchical trees with multi-level pendant structures and nonuniform degree distributions, despite extensive results for simpler families such as paths, stars, cycles, and basic caterpillars. For a simple graph $\mathcal{G}$, the Sombor index is defined as \[ \mathrm{SO}(\mathcal{G}) = \sum_{uv \in E(\mathcal{G})} \sqrt{d(v)^2 + d(u)^2}. \]
In this work, we derive a general recursive formula for the Sombor index of multi-level pendant-augmented path trees. These trees are constructed from a spine path $\mathcal{P}_n$ ($n \ge 2$) in which each vertex has degree $2+k$ and are iteratively augmented over $m \ge 1$ hierarchical levels. Pendants attached to odd-indexed spine vertices branch with replication factor $k$ and terminal degree $\ell_i$, whereas those stemming from even-indexed vertices incorporate an initial offset $\ell_1>2$ that propagates through subsequent levels. These results significantly advance the theoretical and computational study of degree-based topological descriptors in iteratively constructed graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03364 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structural Components Dominate Asymptotic Behavior on Sombor Index with Iterated Pendant Constructions Hamoud, Jasem General Mathematics 05C05, 05C12, 05C20, 05C25, 05C35, 05C76, 68R10 G.2.2 The Sombor index, a degree-based topological descriptor introduced by Gutman in 2021, lacks closed-form expressions for complex hierarchical trees with multi-level pendant structures and nonuniform degree distributions, despite extensive results for simpler families such as paths, stars, cycles, and basic caterpillars. For a simple graph $\mathcal{G}$, the Sombor index is defined as \[ \mathrm{SO}(\mathcal{G}) = \sum_{uv \in E(\mathcal{G})} \sqrt{d(v)^2 + d(u)^2}. \] In this work, we derive a general recursive formula for the Sombor index of multi-level pendant-augmented path trees. These trees are constructed from a spine path $\mathcal{P}_n$ ($n \ge 2$) in which each vertex has degree $2+k$ and are iteratively augmented over $m \ge 1$ hierarchical levels. Pendants attached to odd-indexed spine vertices branch with replication factor $k$ and terminal degree $\ell_i$, whereas those stemming from even-indexed vertices incorporate an initial offset $\ell_1>2$ that propagates through subsequent levels. These results significantly advance the theoretical and computational study of degree-based topological descriptors in iteratively constructed graphs. |
| title | Structural Components Dominate Asymptotic Behavior on Sombor Index with Iterated Pendant Constructions |
| topic | General Mathematics 05C05, 05C12, 05C20, 05C25, 05C35, 05C76, 68R10 G.2.2 |
| url | https://arxiv.org/abs/2603.03364 |