Resistance distance

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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.

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[edit] Definition

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

\Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}\,

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

[edit] Properties of resistance distance

If i=j then

\Omega_{i,j}=0\,.

For an undirected graph

\Omega_{i,j}=\Omega_{j,i}=\Gamma_{i,i}+\Gamma_{j,j}-2\Gamma_{i,j}\,

[edit] General sum rule

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

\sum_{i,j \in V}(LML)_{i,j}\Omega_{i,j}=-2tr(ML)\,

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

\sum_{(i,j) \in E}\Omega_{i,j}=N-1
\sum_{i<j \in V}\Omega_{i,j}=N\sum_{k=1}^{N-1} \lambda_{k}^{-1}

where the λk are the non-zero eigenvalues of the Laplacian matrix.

[edit] 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 by expressed as a function of the set of spanning trees, T, of G as follows:

\Omega_{i,j}=\begin{cases} \frac{\left | \{t:t \in T, e_{i,j} \subseteq T\} \right \vert}{\left | T \right \vert}, & (i,j) \in E\\ \frac{\left | T^'-T \right \vert}{\left | T \right \vert}, &(i,j) \not \in E \end{cases}

where T' is the set of spanning trees for the graph G' = (V,E + ei,j).

[edit] As a squared Euclidean distance

Since the Laplacian L is symmetric and positive semi-definite, it's pseudoinverse Γ is also symmetric and positive semi-definite. Thus, there is a K such that Γ = KKT and we can write:

\Omega_{i,j} = \Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i} = K_iK_i^T + K_jK_j^T - K_iK_j^T - K_jK_i^T = (K_i - K_j)^2

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

[edit] Classical mechanics

In classical mechanics, resistance distance is defined as the distance from the resistance (on a lever) to the fulcrum.

[edit] References