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 resistance of one ohm. It is a metric on graphs.

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

\Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}, $$
 * where $$\Gamma = \left(L + \frac{1}{|V|}\Phi\right)^+,$$

with $+$ denotes the Moore–Penrose inverse, $L$ the Laplacian matrix of $G$, $|V|$ is the number of vertices in $G$, and $Φ$ is the $|V| × |V|$ matrix containing all 1s.

Properties of resistance distance
If $i = j$ then $Ωi,j = 0$. 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;


 * $$\begin{align}

\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} \end{align}$$

where the $λk$ are the non-zero eigenvalues of the Laplacian matrix. This unordered sum
 * $$\sum_{i<j} \Omega_{i,j}$$

is called the Kirchhoff index of the graph.

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



\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 $G' = (V, E + ei,j)$. 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 $$C_{u,v}=2m\Omega_{u,v}$$.

As a squared Euclidean distance
Since the Laplacian $L$ is symmetric and positive semi-definite, so is
 * $$\left(L+\frac{1}{|V|}\Phi\right),$$

thus its pseudo-inverse $Γ$ is also symmetric and positive semi-definite. Thus, there is a $K$ such that $$\Gamma = KK^\textsf{T}$$ and we can write:


 * $$\Omega_{i,j} = \Gamma_{i,i} + \Gamma_{j,j} - \Gamma_{i,j} - \Gamma_{j,i} = K_iK_i^\textsf{T} + K_j K_j^\textsf{T} - K_i K_j^\textsf{T} - K_j K_i^\textsf{T} = \left(K_i - K_j\right)^2$$

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 $i ∈ {1, 2, 3, …, n}$ is
 * $$\frac{ F_{2(n-i)+1} F_{2i-1} }{ F_{2n} }$$

where $Fj$ is the $j$-th Fibonacci number, for $j ≥ 0$.