An Identity on Resistance Distances

Abstract:

On the basis of electrical network theory, Klein and Randić [Journal of Mathematical Chemistry 12 (1993) 81-9 proposed the novel concept of resistance distance. They view a graph as an (resistive) electrical network by considering each edge of the graph as a unit resistor. Then the resistance distance between any two vertices is defined as the effective resistance between these two nodes in the corresponding electrical network. In the present work, an interesting identity on resistance distances is obtained, that is, the sum of the resistance distance between end-vertices of an edge e of a plane graph G and the resistance distance between end-vertices of the dual edge e^* of e in the dual graph G^* of G is equal to one.

• Periodical: Advanced Materials Research (Volume 748)
• Pages: 1024-1027
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.748.1024
• Online since: August 2013
• Authors: Yu Jun Yang
• Keywords: Dual Graph, Plane Graph, Resistance Distance

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