If then . 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 -vertex
simple connected graph and arbitrary
matrix : :\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 . Two of note are; :\begin{align} \sum_{(i,j) \in E}\Omega_{i,j} &= N - 1 \\ \sum_{i where the are the non-zero
eigenvalues of the
Laplacian matrix. This unordered sum :\sum_{i is called the Kirchhoff index of the graph.
Relationship to the number of spanning trees of a graph For a simple connected graph , the
resistance distance between two vertices may be expressed as a
function of the
set of
spanning trees, , of 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 is the set of spanning trees for the graph . 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 v 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}. Resistance distance is also related to the
escape probability between two vertices. The escape probability P_{u,v} between u and v is the probability that a random walk starting at u visits v before returning to u. The escape probability equals : P_{u,v} = \frac{1}{\deg(u)\Omega_{u,v}}, where \deg(u) is the
degree of u. Unlike the commute time, the escape probability is not symmetric in general, i.e., P_{u,v}\neq P_{v,u}.
As a squared Euclidean distance Since the Laplacian is symmetric and positive semi-definite, so is :\left(L+\frac{1}\Phi\right), thus its pseudo-inverse is also symmetric and positive semi-definite. Thus, there is a 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 .
Connection with Fibonacci numbers A fan graph is a graph on vertices where there is an edge between vertex and for all , and there is an edge between vertex and for all . The resistance distance between vertex and vertex {{math|
i ∈ {1, 2, 3, …,
n} }} is :\frac{ F_{2(n-i)+1} F_{2i-1} }{ F_{2n} } where is the -th Fibonacci number, for . == See also ==