On the KG-Sombor index

Introduction/purpose: Degree-based graph invariants are a type of molecular descriptor that represent the connectivity of atoms (vertices) in a molecule through bonds (edges). They are used to model structural properties of molecules and provide valuable information for fields such as physical chemistry, pharmacology, environmental science, and material science. Recently, novel degree-based molecular structure descriptors, known as Sombor index-like graph invariants, have been explored from a geometrical perspective. These graph invariants have found applications in network science, where they are used to model dynamic effects in biological, social, and technological complex systems. There is also emerging interest in their potential military applications. Among these descriptors is the KG-Sombor index which is defined using both vertex and edge degrees. Methods: The study uses combinatorial graph theory to identify and analyze extremal graphs that either maximize or minimize the KG-Sombor index. Results: The extremal graphs are characterized concerning the KG-Sombor index, with a particular focus on trees, molecular trees, and unicyclic graphs. Conclusion: This research advances the theoretical understanding of Sombor index-like graph invariants.