Abstract
The calculation of two-point resistance and the number of spanning trees of a resistor network is a classic problem in electric circuit theory, which has been studied extensively by many mathematicians and physicists for many years. Ge (2021) obtained the formulae for the resistance distances and the number of spanning trees in a complete bipartite graph plus a matching. In this paper, we give a so-called complete multipartite graph-blossomed star transformation which is similar to the mesh-star transformation in electrical networks. As applications, we obtain the close formulae of resistance distances and the number of spanning trees in a complete multipartite graph adding a matching. We also derive the formulae of the resistance distances in the almost complete graph, which is obtained from a complete graph by deleting a forest in which each component is a star.
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