Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes
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arXiv
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2025
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| author | Hamoud, Jasem Abdullah, Duaa |
| author_facet | Hamoud, Jasem Abdullah, Duaa |
| contents | In this paper, the investigates Adriatic indices, specifically the sum lordeg index where it defined as $SL(G) = \sum_{u \in V(G)} °_G(u) \sqrt{\ln °_G(u)}$ and the variable sum exdeg index $SEI_a(G)$ for $a>0$, $a\neq 1$. We present several sharp bounds and characterizations of these and related topological indices on specialized graph classes, including regular graphs, thorny graphs, and chemical trees. Using the strict convexity of function $f$, inequalities for degree-based graph invariants $H_f(T)$ are derived under structural constraints on trees such as branching vertices and maximum degree. Examples on caterpillar trees illustrate the computation of indices like $^{m}M_2(G)$, $F(G)$, $M_2(G)$, and others, revealing the interplay between degree sequences and index values. Additionally, upper and lower bounds on the Sombor index $SO(G^*)$ of thorny graphs $G^*$ are established as \[ \operatorname{SO} \leqslant \sum_{uv\in E(G)}\sqrt{\frac{1}{°_{G}(u)^2+°_{G}(v)^2}+°_{G}(u)+°_{G}(v)}, \]
including criteria for equality, with implications for regular and thorn-regular graphs. The treatment includes detailed formulas, constructive examples, and inequalities critical for understanding the relationship between graph topology and vertex-degree-based descriptors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes Hamoud, Jasem Abdullah, Duaa Combinatorics 05C05, 05C12, 05C20, 05C25, 05C35, 05C76, 68R10 G.2.2 In this paper, the investigates Adriatic indices, specifically the sum lordeg index where it defined as $SL(G) = \sum_{u \in V(G)} °_G(u) \sqrt{\ln °_G(u)}$ and the variable sum exdeg index $SEI_a(G)$ for $a>0$, $a\neq 1$. We present several sharp bounds and characterizations of these and related topological indices on specialized graph classes, including regular graphs, thorny graphs, and chemical trees. Using the strict convexity of function $f$, inequalities for degree-based graph invariants $H_f(T)$ are derived under structural constraints on trees such as branching vertices and maximum degree. Examples on caterpillar trees illustrate the computation of indices like $^{m}M_2(G)$, $F(G)$, $M_2(G)$, and others, revealing the interplay between degree sequences and index values. Additionally, upper and lower bounds on the Sombor index $SO(G^*)$ of thorny graphs $G^*$ are established as \[ \operatorname{SO} \leqslant \sum_{uv\in E(G)}\sqrt{\frac{1}{°_{G}(u)^2+°_{G}(v)^2}+°_{G}(u)+°_{G}(v)}, \] including criteria for equality, with implications for regular and thorn-regular graphs. The treatment includes detailed formulas, constructive examples, and inequalities critical for understanding the relationship between graph topology and vertex-degree-based descriptors. |
| title | Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes |
| topic | Combinatorics 05C05, 05C12, 05C20, 05C25, 05C35, 05C76, 68R10 G.2.2 |
| url | https://arxiv.org/abs/2508.04518 |