Extremal chemical graphs of maximum degree at most 3 for 33 degree-based topological indices
Sébastien Bonte, Gauvain Devillez, Valentin Dusollier, Alain Hertz, Hadrien Mélot
TL;DR
The paper investigates extremal graphs among connected graphs with maximum degree at most $3$ for 33 degree-based topological indices, introducing a compact description via five core graph families. It develops a transform-based toolkit, including $(A,k)$-transforms and preserving vectors, to prove that extremal structures collapse to a small number of patterns across many indices. Specifically, 29 indices have extremal graphs describable by five families, while additional families extend coverage to the remaining indices, revealing substantial commonality in extremal graphs across diverse descriptors. This framework offers a practical method to assess whether a new index shares the extremal structure observed for many established indices.
Abstract
We consider chemical graphs that are defined as connected graphs of maximum degree at most 3. We characterize the extremal graphs, meaning those that maximize or minimize 33 degree-based topological indices. This study shows that five graph families are sufficient to characterize the extremal graphs of 29 of these 33 indices. In other words, the extremal properties of this set of degree-based topological indices vary very little.