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 $SL(G) = \sum_{u \in V(G)} \deg_G(u) \sqrt{\ln \deg_G(u)}$ 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}{\deg_{G}(u)^2+\deg_{G}(v)^2}+\deg_{G}(u)+\deg_{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.