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Resistance distance
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===Relationship to the number of spanning trees of a graph=== For a simple connected graph {{math|1=''G'' = (''V'', ''E'')}}, the '''resistance distance''' between two vertices may be expressed as a [[Function (mathematics)|function]] of the [[Set (mathematics)|set]] of [[spanning tree (mathematics)|spanning trees]], {{mvar|T}}, of {{mvar|G}} as follows: :<math> \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} </math> where {{mvar|T'}} is the set of spanning trees for the graph {{math|1=''G' ''= (''V'', ''E'' + ''e''{{sub|''i'',''j''}})}}. In other words, for an edge <math>(i,j)\in E</math>, the resistance distance between a pair of nodes <math>i</math> and <math>j</math> is the probability that the edge <math>(i,j)</math> is in a random spanning tree of <math>G</math>.
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