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

TL;DR

This paper derives bounds and characterizations for degree-based topological indices on specialized graph classes, enhancing understanding of how graph structure influences these indices, with applications to chemical and thorny graphs.

Contribution

It introduces new sharp bounds and characterizations for Adriatic and related indices on specialized graphs, including regular, thorny, and chemical trees, using convexity and structural constraints.

Findings

Sharp bounds for sum lordeg index on various graph classes

Inequalities for degree-based invariants using convex functions

Bounds on Sombor index for thorny graphs with equality conditions

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.