In this paper, we investigate topological indices, specifically the sum lordeg index and the variable sum edge 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 the 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.