Graph vertices ranking using absolute potentials of electric circuit nodes

A method for ranking the vertices of a graph based on Kirchhoff's laws for determining the potentials of an electrical network is proposed. The graph is represented as an electrical network, where the edge weights are interpreted as electrical conductivities. Next, the current is supplied to one of the vertices, and the absolute potentials of all vertices are determined by the Kirchhoff method. Based on the obtained potential values, the vertices are ranked. Then the current is sequentially applied to all vertices and the ranking is performed each time. For the final ranking, it is proposed to apply the ranking procedure based on the tournament matrix. The operation of the ranking algorithm is illustrated by numerical examples related to graphs of specific transport networks.