Computing the expected value of invariants based on resistance distance for random spiro-polynomio chains

Random polynomio chains adopt a geometrically guided framework for analyzing resistance distance, which involves mathematical techniques to comprehend electrical resistance and optimize communication routes within networks. By determining the expected values of resistance distance-based indices, we can understand the typical or average performance of the network in terms of electrical resistance and communication efficiency. In this study, the closed-form formulae for the expected values of the Kirchhoff and additive degree-Kirchhoff indices for the random spiro-polynomio chains are determined. Furthermore, we compute the average values of Kirchhoff and additive degree-Kirchhoff indices for the spiro-polynomio chains with $n$ polynomios.