Resistance distance

In graph theory, the resistance distance between two vertices of a simple connected graph, G, is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a 1 ohm resistance. It is a metric on graphs.

Definition

On a graph G, the resistance distance Ωi,j between two vertices vi and vj is

where Γ is the Moore–Penrose inverse of the Laplacian matrix of G.

Properties of resistance distance

If i = j then

For an undirected graph

General sum rule

For any N-vertex simple connected graph G = (V, E) and arbitrary N×N matrix M:

From this generalized sum rule a number of relationships can be derived depending on the choice of M. Two of note are;

where the are the non-zero eigenvalues of the Laplacian matrix. This unordered sum Σi<jΩi,j is called the Kirchhoff index of the graph.

Relationship to the number of spanning trees of a graph

For a simple connected graph G = (V, E), the resistance distance between two vertices may be expressed as a function of the set of spanning trees, T, of G as follows:

where is the set of spanning trees for the graph .

As a squared Euclidean distance

Since the Laplacian is symmetric and positive semi-definite, its pseudoinverse is also symmetric and positive semi-definite. Thus, there is a such that and we can write:

showing that the square root of the resistance distance corresponds to the Euclidean distance in the space spanned by .

Connection with Fibonacci numbers

A fan graph is a graph on vertices where there is an edge between vertex and for all and there is an edge between vertex and for all

The resistance distance between vertex and vertex is where is the -th Fibonacci number, for .[1][2]

See also

References

  1. Bapat, R. B.; Gupta, Somit (2010). "Resistance distance in wheels and fans". Indian Journal of Pure and Applied Mathematics. 41: 1–13. CiteSeerX 10.1.1.418.7626. doi:10.1007/s13226-010-0004-2.
  2. http://www.isid.ac.in/~rbb/somitnew.pdf
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