Structural Components Dominate Asymptotic Behavior on Sombor Index with Iterated Pendant Constructions

Abstract

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.

Figures (5)

• Figure 1: Caterpillar trees are precisely those trees characterized by Lemma \ref{['SO002caterpillar']}.
• Figure 2: Augmented path graph corresponding to Lemma \ref{['SO003caterpillar']}. Odd-indexed spine vertices are attached to $k$ pendant vertices of degree $k$, while even-indexed spine vertices are attached to $k$ pendant vertices of degree $k+1$.
• Figure 3: Hierarchical Iterated Unicyclic Graph with $n=7$ (furthermore see Cvetkovi1987).
• Figure 4: Growth of topological indices by comparing Sombor index under iterated pendant extension.
• Figure 5: Growth of topological indices by comparing Sombor index with curve fits.

Theorems & Definitions (26)

• Definition 1: Zagreb Indices gutman1972totalgutman1975acyclic
• Lemma 1.1: Das2021
• Theorem 1.2: Gutman2021Geo
• Theorem 1.3: Gutman2021GeoLiu2023GutmanYou
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.1
• Lemma 2.2