On the square PI index of graphs

Szeged-type indices form a class of distance-based structural descriptors in chemical graph theory, encompassing the Szeged index, the vertex PI index, the Mostar index, the Trinajsti index, and the square PI index (SP I), the last of which was recently introduced by Brezovnik and Tratnik [Int.J.Quantum Chem.121 (2021) #e26530].To date, no systematic study of SP I has been conducted, and the present paper aims to fill this gap.We establish several mathematical properties of SP I, including its relationships with the Szeged index, and we determine the graphs that maximize SP I among cactus graphs of a given order.We conclude the paper by proposing a conjecture and a few open problems.