Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes

Abstract

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 $SE{I}_a(G)$ for $a>0$, $a\ne 1$. 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 $SO(G^{\ast })$ of thorny graphs $G^{\ast }$ 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.

19 pages, 3 tables, Comments welcome!. arXiv admin note: text overlap with arXiv:1209.0275 by other authors