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 Ω_i,j between two vertices v_i and v_j is

$\Omega _{{i,j}}:=\Gamma _{{i,i}}+\Gamma _{{j,j}}-\Gamma _{{i,j}}-\Gamma _{{j,i}}\,$

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

Properties of resistance distance

If i = j then

$\Omega _{{i,j}}=0.\,$

For an undirected graph

$\Omega _{{i,j}}=\Omega _{{j,i}}=\Gamma _{{i,i}}+\Gamma _{{j,j}}-2\Gamma _{{i,j}}\,$

General sum rule

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

$\sum _{{i,j\in V}}(LML)_{{i,j}}\Omega _{{i,j}}=-2\operatorname {tr}(ML)\,$

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 $\lambda _{k}$ are the non-zero eigenvalues of the Laplacian matrix. This unordered sum Σ_i<jΩ_i,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 by expressed as a function of the set of spanning trees, T, of G as follows:

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

where $T'$ is the set of spanning trees for the graph $G'=(V,E+e_{{i,j}})$.

As a squared Euclidean distance

Since the Laplacian $L$ is symmetric and positive semi-definite, its pseudoinverse $\Gamma$ is also symmetric and positive semi-definite. Thus, there is a $K$ such that $\Gamma =KK^{T}$ and we can write:

$\Omega _{{i,j}}=\Gamma _{{i,i}}+\Gamma _{{j,j}}-\Gamma _{{i,j}}-\Gamma _{{j,i}}=K_{i}K_{i}^{T}+K_{j}K_{j}^{T}-K_{i}K_{j}^{T}-K_{j}K_{i}^{T}=(K_{i}-K_{j})^{2}$

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

Connection with Fibonacci numbers

A fan graph is a graph on $n+1$ vertices where there is an edge between vertex $i$ and $n+1$ for all $i = 1, 2, 3, ...n,$ and there is an edge between vertex $i$ and $i+1$ for all $i = 1, 2, 3, ..., n-1.$

The resistance distance between vertex $n+1$ and vertex $i \in \{1,2,3,...,n\}$ is $\frac{ F_{2(n-i)+1} F_{2i-1} }{ F_{2n} }$ where $F_{j}$ is the $j$-th Fibonacci number, for $j\geq 0$.^[1][2]

See also

• Conductance (graph)

References

1. "Resistance distance in wheels and fans". Indian Journal of Pure and Applied Mathematics. 41: 1–13. doi:10.1007/s13226-010-0004-2.
2. http://www.isid.ac.in/~rbb/somitnew.pdf

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