Exploring Bounds on Generalized Szeged Indices of Graphs

Abstract

Topological indices are graph-based numerical descriptors that reflect key structural properties such as connectivity, branching, and distance distribution. Two classical and widely studied distance-based indices are the Szeged index and the PI (Peripheral) index. The Szeged index, introduced by Gutman [1], partitions vertex sets based on shortest-path distances relative to edges, providing insights into the branching nature of molecular graphs. In contrast, the general PI index [2] quantifies the imbalance in distances from each vertex to the endpoints of an edge, offering a complementary perspective on molecular shape. These indices have inspired several refinements to improve sensitivity and applicability. Das et al. [3] extended the Szeged index to accommodate weighted graphs, enhancing its adaptability to real-world data structures. Klavžar and Nadjafi-Arani [4] developed a decomposition-based approach to simplify the computation of the index for large-scale graphs. These advancements demonstrate the ongoing significance of Szeged-type indices in mathematical and applied graph theory.