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| '''Colin de Verdière's invariant''' is a graph parameter <math>\mu(G)</math> for any [[Graph (mathematics)|graph]] ''G,'' introduced by [[Yves Colin de Verdière]] in 1990. It was motivated by the study of the maximum multiplicity of the second [[eigenvalue]] of certain [[Schrödinger operator]]s.<ref name="hls99"/>
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| ==Definition==
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| Let <math>G=(V,E)</math> be a loopless simple graph. Assume without loss of generality that <math>V=\{1,\dots,n\}</math>. Then <math>\mu(G)</math> is the largest [[corank]] of any [[symmetric matrix]] <math>M=(M_{i,j})\in\mathbb{R}^{(n)}</math> such that:
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| * (M1) for all <math>i,j</math> with <math>i\neq j</math>: <math>M_{i,j}<0</math> if ''i'' and ''j'' are adjacent, and <math>M_{i,j}=0</math> if ''i'' and ''j'' are nonadjacent;
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| * (M2) ''M'' has exactly one negative eigenvalue, of multiplicity 1;
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| * (M3) there is no nonzero matrix <math>X=(X_{i,j})\in\mathbb{R}^{(n)}</math> such that <math>MX=0</math> and such that <math>X_{i,j}=0</math> whenever <math>i=j</math> or <math>M_{i,j}\neq 0</math>.<ref name="hls99"/><ref name="cdv90"/>
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| ==Characterization of known graph families==
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| Several well-known families of graphs can be characterized in terms of their Colin de Verdière invariants:
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| *{{nowrap|μ ≤ 0}} if and only if ''G'' has [[empty graph|no edges]];<ref name="hls99">{{harvtxt|van der Holst|Lovász|Schrijver|1999}}.</ref><ref name="cdv90"/>
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| *{{nowrap|μ ≤ 1}} if and only if ''G'' is a [[linear forest]] (disjoint union of paths);<ref name="hls99"/><ref>{{harvtxt|Colin de Verdière|1990}} does not state this case explicitly, but it follows from his characterization of these graphs as the graphs with no [[triangle graph]] or [[claw (graph theory)|claw]] minor.</ref>
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| *{{nowrap|μ ≤ 2}} if and only if ''G'' is [[outerplanar graph|outerplanar]];<ref name="hls99"/><ref name="cdv90"/>
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| *{{nowrap|μ ≤ 3}} if and only if ''G'' is [[planar graph|planar]];<ref name="hls99"/><ref name="cdv90">{{harvtxt|Colin de Verdière|1990}}.</ref>
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| *{{nowrap|μ ≤ 4}} if and only if ''G'' is [[linkless embedding|linklessly embeddable graph]]<ref name="hls99"/><ref name="ls98">{{harvtxt|Lovász|Schrijver|1998}}.</ref>
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| These same families of graphs also show up in connections between the Colin de Verdière invariant of a graph and the structure of its [[complement graph]]:
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| *If the complement of an ''n''-vertex graph is a linear forest, then {{nowrap|μ ≥ ''n'' − 3}};<ref name="hls99"/><ref name="klv97">{{harvtxt|Kotlov|Lovász|Vempala|1997}}.</ref>
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| *If the complement of an ''n''-vertex graph is outerplanar, then {{nowrap|μ ≥ ''n'' − 4}};<ref name="hls99"/><ref name="klv97"/>
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| *If the complement of an ''n''-vertex graph is planar, then {{nowrap|μ ≥ ''n'' − 5}}.<ref name="hls99"/><ref name="klv97"/>
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| ==Graph minors==
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| A [[Minor (graph theory)|minor]] of a graph is another graph formed from it by contracting edges and by deleting edges and vertices. The Colin de Verdière invariant is minor-monotone, meaning that taking a minor of a graph can only decrease or leave unchanged its invariant:
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| :If ''H'' is a minor of ''G'' then <math>\mu(H)\leq\mu(G)</math>.<ref name="cdv90"/>
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| By the [[Robertson–Seymour theorem]], for every ''k'' there exists a finite set ''H'' of graphs such that the graphs with invariant at most ''k'' are the same as the graphs that do not have any member of ''H'' as a minor. {{harvtxt|Colin de Verdière|1990}} lists these sets of [[forbidden minor]]s for ''k'' ≤ 3; for ''k'' = 4 the set of forbidden minors consists of the seven graphs in the [[Petersen family]], due to the two characterizations of the [[linkless embedding|linklessly embeddable graph]]s as the graphs with μ ≤ 4 and as the graphs with no Petersen family minor.<ref name="ls98"/>
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| ==Chromatic number==
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| {{harvtxt|Colin de Verdière|1990}} conjectured that any graph with Colin de Verdière invariant μ may be [[graph coloring|colored]] with at most μ + 1 colors. For instance, the linear forests have invariant 1, and can be [[bipartite graph|2-colored]]; the [[outerplanar graph]]s have invariant two, and can be 3-colored; the [[planar graph]]s have invariant 3, and (by the [[four color theorem]]) can be 4-colored.
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| For graphs with Colin de Verdière invariant at most four, the conjecture remains true; these are the [[linkless embedding|linklessly embeddable graph]]s, and the fact that they have chromatic number at most five is a consequence of a proof by {{harvtxt|Robertson|Seymour|Thomas|1993}} of the [[Hadwiger conjecture (graph theory)|Hadwiger conjecture]] for ''K''<sub>6</sub>-minor-free graphs.
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| ==Other properties==
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| If a graph has [[crossing number (graph theory)|crossing number]] ''k'', it has Colin de Verdière invariant at most ''k'' + 3. For instance, the two Kuratowski graphs ''K''<sub>5</sub> and ''K''<sub>3,3</sub> can both be drawn with a single crossing, and have Colin de Verdière invariant at most four.<ref name="cdv90"/>
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| ==Influence==
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| Colin de Verdière invariant is defined from a special class of matrices corresponding to a graph instead of just a single matrix related to the graph. Along the same line other graph parameters are defined and studied such as [[Minimum rank of a graph]], [[Minimum semidefinite rank of a graph]] and [[Minimum skew rank of a graph]].
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{citation
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| | last = Colin de Verdière | first = Y. | author-link = Yves Colin de Verdière
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| | doi = 10.1016/0095-8956(90)90093-F
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| | issue = 1
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| | journal = [[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]]
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| | pages = 11–21
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| | title = Sur un nouvel invariant des graphes et un critère de planarité
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| | volume = 50
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| | year = 1990}}. Translated by Neil Calkin as {{citation
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| | last = Colin de Verdière | first = Y. | author-link = Yves Colin de Verdière
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| | contribution = On a new graph invariant and a criterion for planarity
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| | editor1-last = Robertson | editor1-first = Neil | editor1-link = Neil Robertson (mathematician)
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| | editor2-last = Seymour | editor2-first = Paul | editor2-link = Paul Seymour (mathematician)
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| | pages = 137–147
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| | publisher = American Mathematical Society
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| | series = Contemporary Mathematics
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| | title = Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors
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| | volume = 147
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| | year = 1993}}.
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| *{{citation
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| | last1 = van der Holst | first1 = Hein
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| | last2 = Lovász | first2 = László | author2-link = László Lovász
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| | last3 = Schrijver | first3 = Alexander | author3-link = Alexander Schrijver
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| | contribution = The Colin de Verdière graph parameter
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| | location = Budapest
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| | pages = 29–85
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| | publisher = János Bolyai Math. Soc.
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| | series = Bolyai Soc. Math. Stud.
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| | title = Graph Theory and Combinatorial Biology (Balatonlelle, 1996)
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| | url = http://www.cs.elte.hu/~lovasz/colinsurv.ps
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| | volume = 7
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| | year = 1999}}.
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| *{{citation
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| | last1 = Kotlov | first1 = Andrew
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| | last2 = Lovász | first2 = László | author2-link = László Lovász
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| | last3 = Vempala | first3 = Santosh
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| | doi = 10.1007/BF01195002
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| | issue = 4
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| | journal = Combinatorica
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| | pages = 483–521
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| | title = The Colin de Verdiere number and sphere representations of a graph
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| | url = http://oldwww.cs.elte.hu/~lovasz/sphere.ps
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| | volume = 17
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| | year = 1997}}
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| *{{citation
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| | last1 = Lovász | first1 = László | author1-link = László Lovász
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| | last2 = Schrijver | first2 = Alexander | author2-link = Alexander Schrijver
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| | doi = 10.1090/S0002-9939-98-04244-0
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| | issue = 5
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| | journal = [[Proceedings of the American Mathematical Society]]
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| | pages = 1275–1285
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| | title = A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
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| | volume = 126
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| | year = 1998}}.
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| *{{citation | last1=Robertson | first1=Neil | author1-link=Neil Robertson (mathematician) | last2=Seymour | first2=Paul | author2-link=Paul Seymour (mathematician) | last3=Thomas | first3=Robin | author3-link=Robin Thomas (mathematician) | title=Hadwiger's conjecture for K<sub>6</sub>-free graphs | url=http://www.math.gatech.edu/~thomas/PAP/hadwiger.pdf | year=1993 | journal=[[Combinatorica]] | volume=13 | pages=279–361 | doi=10.1007/BF01202354}}.
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| {{DEFAULTSORT:Colin de Verdiere graph invariant}}
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| [[Category:Graph invariants]]
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| [[Category:Graph minor theory]]
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The writer's title is Christy Brookins. I am truly fond of to go to karaoke but I've been using on new issues lately. For many years he's been living in Mississippi and he doesn't plan on altering it. Credit authorising is how she makes a residing.
Also visit my website ... tarot card readings; netwk.hannam.ac.kr,