Mathematical principle of m×n resistor networks

he unified processing and research of multiple network models are implemented, and a new theoretical breakthrough is made, which sets up two new theorems on evaluating the exact electrical characteristics (potential and resistance) of the complex m×n resistor networks by the Recursion-Transform method with potential parameters (RT-V), applies to a variety of different types of lattice structure with arbitrary boundaries such as the nonregular m×n rectangular networks and the nonregular m×n cylindrical networks. Our research gives the analytical solutions of electrical characteristics of the complex networks (finite, semi-infinite and infinite), which has not been solved before. As applications of the theorems, a series of analytical solutions of potential and resistance of the complex resistor networks are discovered. In particular, three novel mathematical propositions are discovered when comparing the resistance in two resistor networks, and many interesting trigonometric identities are discovered as well.