Abstract
The classical distance between two vertices of a graph, which is defined to be the length of the shortest path, is often intuitively not appealing, and is also not tractable mathematically. The resistance distance between two vertices is defined to be the effective resistance between the vertices, when the graph is viewed as an electrical network, with a unit resistance along each edge. We begin by giving an equivalent definition in terms of generalized inverse, and prove some basic properties, including the triangle inequality. In the next few sections, interpretaion of the resistance distance in terms of network flows, random walk on the graph and electrical networks is provided. In the final section some properties of the resistance matrix are proved. A formula for the inverse of the resistance matrix is obtained, generalizing the formula for the inverse of the distance matrix of a tree.
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References and Further Reading
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© 2014 Springer-Verlag London
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Bapat, R.B. (2014). Distance Matrix of a Tree. In: Graphs and Matrices. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6569-9_9
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DOI: https://doi.org/10.1007/978-1-4471-6569-9_9
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