|
|
| Line 1: |
Line 1: |
| In [[graph theory]], the '''resistance distance''' between two [[vertex (graph theory)|vertices]] of a [[graph (mathematics)|simple connected graph]], ''G'', is equal to the [[electrical resistance|resistance]] between two equivalent points on an [[electrical network]], constructed so as to correspond to ''G'', with each [[graph (mathematics)|edge]] being replaced by a 1 [[ohm]] [[electrical resistance|resistance]]. It is a [[metric (mathematics)|metric]] on [[graph (mathematics)|graphs]].
| | Anybody who wrote the guide is called Eusebio. His friends say it's negative for him but the text he loves doing is acting and he's has been doing it for much too long. Filing has been his profession as news got around. [http://Search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Massachusetts&Submit=Go Massachusetts] has always been his everyday living place and his family loves it. Go to his website to identify a out more: http://circuspartypanama.com<br><br>My homepage ... hack clash of clans - [http://circuspartypanama.com his response] - |
| | |
| ==Definition==
| |
| | |
| On a [[graph (mathematics)|graph]] ''G'', the '''resistance distance''' Ω<sub>''i'',''j''</sub> between two vertices ''v<sub>i</sub>'' and ''v<sub>j</sub>'' is
| |
| | |
| :<math>
| |
| \Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}\,
| |
| </math>
| |
| | |
| where Γ is the [[Moore–Penrose inverse]] of the [[Laplacian matrix]] of ''G''.
| |
| | |
| ==Properties of resistance distance==
| |
| | |
| If ''i'' = ''j'' then
| |
| | |
| :<math>\Omega_{i,j}=0.\,</math>
| |
| | |
| 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 ''N''-vertex [[graph (mathematics)|simple connected graph]] ''G'' = (''V'', ''E'') and arbitrary ''N''×''N'' [[matrix (mathematics)|matrix]] ''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 ''M''. Two of note are;
| |
| | |
| :<math>\sum_{(i,j) \in E}\Omega_{i,j}=N-1</math>
| |
| | |
| :<math>\sum_{i<j \in V}\Omega_{i,j}=N\sum_{k=1}^{N-1} \lambda_{k}^{-1}</math>
| |
| | |
| where the <math>\lambda_{k}</math> are the non-zero [[eigenvalues]] of the [[Laplacian matrix]]. This unordered sum {{math|Σ<sub>i<j</sub>Ω</sub>i,j</sub>}} 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 by expressed as a [[Function (mathematics)|function]] of the [[Set (mathematics)|set]] of [[spanning tree (mathematics)|spanning trees]], ''T'', of ''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 <math>T'</math> is the set of spanning trees for the graph <math>G'=(V, E+e_{i,j})</math>.
| |
| | |
| ===As a squared Euclidean distance===
| |
| Since the Laplacian <math>L</math> is symmetric and positive semi-definite, its pseudoinverse <math>\Gamma</math> is also symmetric and positive semi-definite. Thus, there is a <math>K</math> such that <math>\Gamma = K K^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^T + K_jK_j^T - K_iK_j^T - K_jK_i^T = (K_i - K_j)^2</math> | |
| | |
| showing that the square root of the resistance distance corresponds to the [[Euclidean distance]] in the space spanned by <math>K</math>.
| |
| | |
| == See also ==
| |
| * [[Conductance (graph)]]
| |
| | |
| ==References==
| |
| * {{cite journal
| |
| |first1= D. J.
| |
| |last1=Klein
| |
| |first2=M. J.
| |
| |last2=Randic
| |
| |journal=J. Math. Chem.
| |
| |doi=10.1007/BF01164627
| |
| |title=Resistance Distance
| |
| |volume=12
| |
| |page=81
| |
| |year=1993
| |
| }}
| |
| * {{cite journal
| |
| |first1=Ivan
| |
| |last1=Gutman
| |
| |first2=Bojan
| |
| |last2=Mohar
| |
| |title=The quasi-Wiener and the Kirchhoff indices coincide
| |
| |year=1996
| |
| |journal=J. Chem. Inf. Comput. Sci.
| |
| |volume=36
| |
| |pages=982–985
| |
| |doi=10.1021/ci960007t
| |
| }}
| |
| * {{cite journal
| |
| |title=Closed-form formulas for the Kirchhoff index
| |
| |first1=Jose Luis
| |
| |last1=Palacios
| |
| |journal=Int. J. Quant. Chem.
| |
| |year=2001
| |
| |volume=81
| |
| |pages=135–140
| |
| |doi=10.1002/1097-461X(2001)81:2<135::AID-QUA4>3.0.CO;2-G
| |
| }}
| |
| * {{cite journal
| |
| |first1=D.
| |
| |last1=Babic
| |
| |first2=D. J.
| |
| |last2=Klein
| |
| |first3=I.
| |
| |last3=Lukovits
| |
| |first4=S.
| |
| |last4=Nikolic
| |
| |first5=N.
| |
| |last5=Trinajstic
| |
| |title=Resistance-distance matrix: a computational algorithm and its application
| |
| |journal=Int. J. Quant. Chem.
| |
| |year=2002
| |
| |doi=10.1002/qua.10057
| |
| |volume=90
| |
| |number=1
| |
| |pages=166–167
| |
| }}
| |
| * {{cite journal
| |
| |url=http://www.stkpula.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf
| |
| |journal=Croatica Chem. Acta
| |
| |last1= Klein
| |
| |first1=D. J.
| |
| |title=Resistance Distance Sum Rules
| |
| |year=2002
| |
| |volume=75
| |
| |number=2
| |
| |pages=633–649
| |
| }}
| |
| * {{cite journal
| |
| |first1=Ravindra B.
| |
| |last1=Bapat
| |
| |first2=Ivan
| |
| |last2=Gutman
| |
| |first3=Wenjun
| |
| |last3=Xiao
| |
| |title=A simple method for computing resistance distance
| |
| |journal=Z. Naturforsch.
| |
| |year=2003
| |
| |volume=58a
| |
| |pages=494–498
| |
| |url=http://www.znaturforsch.com/aa/v58a/c58a.htm#No9
| |
| }}
| |
| * {{cite journal
| |
| |first1=Jose Luis
| |
| |last1=Placios
| |
| |title=Foster's formulas via probability and the Kirchhoff index
| |
| |journal=Method. Comput. Appl. Probal.
| |
| |year=2004
| |
| |volume=6
| |
| |number=4
| |
| |pages=381–387
| |
| |doi=10.1023/B:MCAP.0000045086.76839.54
| |
| }}
| |
| * {{cite journal
| |
| |first1=Enrique
| |
| |last1=Bendito
| |
| |first2=Angeles
| |
| |last2=Carmona
| |
| |first3=Andres M.
| |
| |last3=Encinas
| |
| |first4=Jose M.
| |
| |last4=Gesto
| |
| |title=A formula for the Kirhhoff index
| |
| |journal=Int. J. Quant. Chem.
| |
| |year=2008
| |
| |doi=10.1002/qua.21588
| |
| |pages=1200–1206
| |
| |volume=108
| |
| |number=6
| |
| |bibcode = 2008IJQC..108.1200B }}
| |
| * {{cite journal
| |
| |journal=Int. J. Quant. Chem.
| |
| |first1=Bo
| |
| |last1=Zhou
| |
| |title=The Kirchhoff index and the matching number
| |
| |first2=Nenad
| |
| |last2=Trinajstic
| |
| |volume=109
| |
| |year=2009
| |
| |pages=2978–2981
| |
| |doi=10.1002/qua.21915
| |
| |bibcode = 2009IJQC..109.2978Z }}
| |
| * {{cite journal
| |
| |first1=Bo
| |
| |last1=Zhou
| |
| |first2=Nenad
| |
| |last2=Trinajstic
| |
| |title=On resistance-distance and the Kirchhoff index
| |
| |journal=J. Math. Chem.
| |
| |year=2009
| |
| |volume=46
| |
| |pages=283–289
| |
| |doi=10.1007/s10910-008-9459-3
| |
| }}
| |
| * {{cite arxiv
| |
| |first1=Bo
| |
| |last1=Zhou
| |
| |title=On sum of powers of Laplacian eigenvalues and Laplacian Estrada Index of graphs
| |
| |eprint=1102.1144
| |
| }}
| |
| | |
| * {{cite journal
| |
| |first1=Heping
| |
| |last1=Zhang
| |
| |first2=Yujun
| |
| |last2=Yang
| |
| |title=Resistance distance and Kirchhoff index in circulant graphs
| |
| |journal=Int. J. Quantum Chem.
| |
| |year=2007
| |
| |volume=107
| |
| |pages=330–339
| |
| |doi=10.1002/qua.21068
| |
| |bibcode = 2007IJQC..107..330Z }}
| |
| | |
| * {{cite journal
| |
| |first1=Yujun
| |
| |last1=Yang
| |
| |first2=Heping
| |
| |last2=Zhang
| |
| |title=Some rules on resistance distance with applications
| |
| |journal=J. Phys. A: Math. Theor.
| |
| |year=2008
| |
| |volume=41
| |
| |pages=445203
| |
| |doi=10.1088/1751-8113/41/44/445203
| |
| |bibcode = 2008JPhA...41R5203Y }}
| |
| [[Category:Graph theory]]
| |
Anybody who wrote the guide is called Eusebio. His friends say it's negative for him but the text he loves doing is acting and he's has been doing it for much too long. Filing has been his profession as news got around. Massachusetts has always been his everyday living place and his family loves it. Go to his website to identify a out more: http://circuspartypanama.com
My homepage ... hack clash of clans - his response -