Counting Spanning Trees in Various Products of Two Complete Bipartite Graphs

Calculating the number of spanning trees in a graph is a crucial problem in combinatorics and physics that has been thoroughly researched for many years by mathematicians and physicists. The higher the quality and perfection of the network, the greater the number of trees spanning it, leading to greater possibilities for connection between two vertices and ensuring good rigidity and resistance. In this work, we use matrix theory and linear algebra techniques to derive simple explicit formulas for calculating the complexity of various products of two complete bipartite graphs, including the Cartesian product, tensor product, normal product, composition product, symmetric product, disjunction, and strong sum.