View source for Resistance distance

{{short description|Graph metric of electrical resistance between nodes}}

In [[graph theory]], the '''resistance distance''' between two [[vertex (graph theory)|vertices]] of a [[Graph (discrete mathematics)|simple]], [[connected graph]], {{mvar|G}}, is equal to the [[electrical resistance|resistance]] between two equivalent points on an [[electrical network]], constructed so as to correspond to {{mvar|G}}, with each [[graph (discrete mathematics)|edge]] being replaced by a [[electrical resistance|resistance]] of one [[ohm]]. It is a [[metric (mathematics)|metric]] on [[Graph (discrete mathematics)|graphs]].

==Definition==

On a [[Graph (discrete mathematics)|graph]] {{mvar|G}}, the '''resistance distance''' {{math|Ω{{sub|''i'',''j''}}}}  between two vertices {{mvar|v{{sub|i}}}} and {{mvar|v{{sub|j}}}} is<ref>{{Cite web|url=https://mathworld.wolfram.com/ResistanceDistance.html|title = Resistance Distance}}</ref>
:<math>
\Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i},
</math>
:where <math>\Gamma = \left(L + \frac{1}{|V|}\Phi\right)^+,</math>
with {{math|{{sup|+}}}} denotes the [[Moore–Penrose inverse]], {{mvar|L}} the [[Laplacian matrix]] of {{mvar|G}}, {{math|{{abs|''V''}}}} is the number of vertices in {{mvar|G}}, and {{math|Φ}} is the {{math|{{abs|''V''}} × {{abs|''V''}}}} matrix containing all 1s.

==Properties of resistance distance==

If {{math|1=''i'' = ''j''}} then {{math|1=Ω{{sub|''i'',''j''}} = 0}}. For an undirected graph
:<math>\Omega_{i,j}=\Omega_{j,i}=\Gamma_{i,i}+\Gamma_{j,j}-2\Gamma_{i,j}</math>

===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.

===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>.

===Relationship to random walks===

The resistance distance between vertices <math>u</math> and <math>u</math> is proportional to the '''commute time''' <math>C_{u,v}</math> of a [[random walk]] between <math>u</math> and <math>v</math>. The commute time is the expected number of steps in a random walk that starts at <math>u</math>, visits <math>v</math>, and returns to <math>u</math>. For a graph with <math>m</math> edges, the resistance distance and commute time are related as <math>C_{u,v}=2m\Omega_{u,v}</math>.<ref>{{cite book |last1=Chandra, Ashok K and Raghavan, Prabhakar and Ruzzo, Walter L and Smolensky, Roman |title=Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89 |chapter=The electrical resistance of a graph captures its commute and cover times |date=1989 |issue=21 |pages=574–685 |doi=10.1145/73007.73062 |isbn=0897913078 |chapter-url=https://dl.acm.org/doi/abs/10.1145/73007.73062}}</ref>

===As a squared Euclidean distance===
Since the Laplacian {{mvar|L}} is symmetric and positive semi-definite, so is 
:<math>\left(L+\frac{1}{|V|}\Phi\right),</math> 
thus its pseudo-inverse {{math|Γ}} is also symmetric and positive semi-definite.  Thus, there is a {{mvar|K}} such that <math>\Gamma = KK^\textsf{T}</math> and we can write:

:<math>\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</math>

showing that the square root of the resistance distance corresponds to the [[Euclidean distance]] in the space spanned by {{mvar|K}}.

===Connection with Fibonacci numbers ===
A fan graph is a graph on {{math|''n'' + 1}} vertices where there is an edge between vertex {{mvar|i}} and {{math|''n'' + 1}} for all {{math|1=''i'' = 1, 2, 3, …, ''n''}}, and there is an edge between vertex {{mvar|i}} and {{mvar|''i'' + 1}} for all {{math|1=i = 1, 2, 3, …, ''n'' – 1}}.

The resistance distance between vertex {{math|''n'' + 1}} and vertex {{math|''i'' ∈ {1, 2, 3, …, ''n''} }} is 
:<math>\frac{ F_{2(n-i)+1} F_{2i-1} }{ F_{2n} }</math> 
where {{mvar|F{{sub|j}}}} is the {{mvar|j}}-th Fibonacci number, for {{math|''j'' ≥ 0}}.<ref>{{cite journal | doi=10.1007/s13226-010-0004-2 | volume=41 | title=Resistance distance in wheels and fans | journal=Indian Journal of Pure and Applied Mathematics | pages=1–13|year = 2010|last1 = Bapat|first1 = R. B.| last2=Gupta | first2=Somit | citeseerx=10.1.1.418.7626 | s2cid=14807374 }}</ref><ref>http://www.isid.ac.in/~rbb/somitnew.pdf {{Bare URL PDF|date=March 2022}}</ref>

== See also ==
* [[Conductance (graph)]]

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[[Category:Electrical resistance and conductance]]
[[Category:Graph distance]]