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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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