University of Mohaghegh Ardabili Journal of Hyperstructures 2251-8436 12 2 2023 12 01 On some topological indices over rectangular grids 322 335 2806 10.22098/jhs.2023.2806 EN Hadi Parsian Department of Civil Engineering, Faculty of Civil Engineering, Art and Architecture, WTIAU, Tehran, Iran. Ali Parsian Department of Mathematics, Tafresh University, Tafresh 39518 79611, Iran Journal Article 2023 11 03 A topological index is a real number related to a graph,which is considered as a structural invariant. Some examples are<br />Sombor index, Randi´c index, Zagreb indices, and Harmonic index.In the present paper, we consider the function Ind from the set of all rectangular grids to the set of real numbers, which assigns to each rectangular grid, one of its above indices. Then we show that the only non-degenerate indices over retangular grids, are Sombor index and Randi´c index, while Zagreb indices and the Harmonic index are degenerate. In the following, we determine rectangular grids with fixed diameter d, where maximum and minimum of the above indices occures on them, in the case m ≥ 3, n ≥ 3. Finally, we find the amounts of these indices.  https://jhs.uma.ac.ir/article_2806_be935c5e9581efc15dd26102146ad0f9.pdf