General sum rule

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

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

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

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

where the λ_k are the non-zero eigenvalues of the Laplacian matrix. This unordered sum

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

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:

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

where T' is the set of spanning trees for the graph G' = (V, E + e_i,j). In other words, for an edge ${\displaystyle (i,j)\in E}$, the resistance distance between a pair of nodes ${\displaystyle i}$ and ${\displaystyle j}$ is the probability that the edge ${\displaystyle (i,j)}$ is in a random spanning tree of ${\displaystyle G}$.

Relationship to random walks

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

As a squared Euclidean distance

Since the Laplacian L is symmetric and positive semi-definite, so is

${\displaystyle \left(L+{\frac {1}{|V|}}\Phi \right),}$

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

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

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

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

where F_j is the j-th Fibonacci number, for j ≥ 0.^[3][4]