Resistance distance

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

Definition

On a graph Template:Mvar, the resistance distance Template:Math between two vertices Template:Mvar and Template:Mvar is

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

where $\mathrm{\Gamma }={(L+\frac{1}{|V|}\mathrm{\Phi })}^{+},$

with Template:Math denotes the Moore–Penrose inverse, Template:Mvar the Laplacian matrix of Template:Mvar, Template:Math is the number of vertices in Template:Mvar, and Template:Math is the Template:Math matrix containing all 1s.

Properties of resistance distance

If Template:Math then Template:Math. For an undirected graph

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

General sum rule

For any Template:Mvar-vertex simple connected graph Template:Math and arbitrary Template:Math matrix Template:Mvar:

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

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

$\begin{array}{cc}\sum _{(i,j)\in E}{\mathrm{\Omega }}_{i,j}& =N-1\\ \sum _{i<j\in V}{\mathrm{\Omega }}_{i,j}& =N\sum _{k=1}^{N-1}{\lambda }_k^{-1}\end{array}$

where the Template:Mvar are the non-zero eigenvalues of the Laplacian matrix. This unordered sum

$\sum _{i<j}{\mathrm{\Omega }}_{i,j}$

is called the Kirchhoff index of the graph.

Relationship to the number of spanning trees of a graph

For a simple connected graph Template:Math, the resistance distance between two vertices may be expressed as a function of the set of spanning trees, Template:Mvar, of Template:Mvar as follows:

${\mathrm{\Omega }}_{i,j}=\{\begin{array}{cc}\frac{|\{t:t\in T,e_{i,j}\in t\}|}{\left|T\right|},& (i,j)\in E\\ \frac{|T^{\prime }-T|}{\left|T\right|},& (i,j)\in ̸E\end{array}$

where Template:Mvar is the set of spanning trees for the graph Template:Math. In other words, for an edge $(i,j)\in E$, the resistance distance between a pair of nodes $i$ and $j$ is the probability that the edge $(i,j)$ is in a random spanning tree of $G$.

Relationship to random walks

The resistance distance between vertices $u$ and $u$ is proportional to the commute time $C_{u,v}$ of a random walk between $u$ and $v$. The commute time is the expected number of steps in a random walk that starts at $u$, visits $v$, and returns to $u$. For a graph with $m$ edges, the resistance distance and commute time are related as $C_{u,v}=2m{\mathrm{\Omega }}_{u,v}$.

As a squared Euclidean distance

Since the Laplacian Template:Mvar is symmetric and positive semi-definite, so is

$(L+\frac{1}{|V|}\mathrm{\Phi }),$

thus its pseudo-inverse Template:Math is also symmetric and positive semi-definite. Thus, there is a Template:Mvar such that $\mathrm{\Gamma }=K{K}^{\mathsf{\text{T}}}$ and we can write:

${\mathrm{\Omega }}_{i,j}={\mathrm{\Gamma }}_{i,i}+{\mathrm{\Gamma }}_{j,j}-{\mathrm{\Gamma }}_{i,j}-{\mathrm{\Gamma }}_{j,i}=K_i{K}_i^{\mathsf{\text{T}}}+K_j{K}_j^{\mathsf{\text{T}}}-K_i{K}_j^{\mathsf{\text{T}}}-K_j{K}_i^{\mathsf{\text{T}}}={(K_i-K_j)}^2$

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

Connection with Fibonacci numbers

A fan graph is a graph on Template:Math vertices where there is an edge between vertex Template:Mvar and Template:Math for all Template:Math, and there is an edge between vertex Template:Mvar and Template:Mvar for all Template:Math.

The resistance distance between vertex Template:Math and vertex Template:Math is

$\frac{F_{2(n-i)+1}{F}_{2i-1}}{F_{2n}}$

where Template:Mvar is the Template:Mvar-th Fibonacci number, for Template:Math.

See also

• Conductance (graph)

References

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