Proposition 6. Given P C n with n ≥ 5 squares. Then Hence, combining this with the results detailed in the previous propositions, the ascending and descending sequences in both indices can be connected for n ≥ 5. On the other hand, when comparing Proposition 5 with Proposition 6, it is noted that the latter includes an additional inequality. This is primarily attributed to the fact that the function g 2 involves four variables, whereas the function g 1 comprises three. In fact, as per the findings presented in , the recursive formula for the degree-based topological indices on the P C n is governed by a function of two variables. Consequently, the question arises as to when similar behaviors persist and when distinct patterns emerge for different degree-based topological indices. Finally, it is crucial to note, although it is not explicitly stated in the above propositions, that due to symmetry )) = P C(2, 2, . . . , 2