Editing Resistance distance (section)

===General sum rule===
For any {{mvar|N}}-vertex [[graph (discrete mathematics)|simple connected graph]] {{math|1=''G'' = (''V'', ''E'')}} and arbitrary {{math|''N''×''N''}} [[matrix (mathematics)|matrix]] {{mvar|M}}:

:<math>\sum_{i,j \in V}(LML)_{i,j}\Omega_{i,j} = -2\operatorname{tr}(ML)</math>

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

:<math>\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}</math>

where the {{mvar|λ{{sub|k}}}} are the non-zero [[eigenvalues]] of the [[Laplacian matrix]]. This unordered sum 
:<math>\sum_{i<j} \Omega_{i,j}</math>
is called the Kirchhoff index of the graph.