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Colin de Verdière's invariant is a graph parameter $\mu (G)$ for any 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 operators.^[1]

Definition

Let $G=(V,E)$ be a simple graph with vertex set $V=\{1,\dots ,n\}$ . Then $\mu (G)$ is the largest corank of any symmetric matrix $M=(M_{i,j})\in \mathbb {R} ^{(n)}$ such that:

(M1) for all $i,j$ with $i\neq j$ : $M_{i,j}<0$ if $\{i,j\}\in E$ , and $M_{i,j}=0$ if $\{i,j\}\notin E$ ;
(M2) $M$ has exactly one negative eigenvalue, of multiplicity 1;
(M3) there is no nonzero matrix $X=(X_{i,j})\in \mathbb {R} ^{(n)}$ such that $MX=0$ and such that $X_{i,j}=0$ if either $i=j$ or $M_{i,j}\neq 0$ hold.^[1]^[2]

Characterization of known graph families

Several well-known families of graphs can be characterized in terms of their Colin de Verdière invariants:

μ ≤ 0 if and only if G has only one vertex;^[1]^[2]
μ ≤ 1 if and only if G is a linear forest (a disjoint union of paths);^[1]^[3]
μ ≤ 2 if and only if G is outerplanar;^[1]^[2]
μ ≤ 3 if and only if G is planar;^[1]^[2]
μ ≤ 4 if and only if G is linklessly embeddable in R³.^[1]^[4]

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:

If the complement of an n-vertex graph is a linear forest, then μ ≥ n − 3;^[1]^[5]
If the complement of an n-vertex graph is outerplanar, then μ ≥ n − 4;^[1]^[5]
If the complement of an n-vertex graph is planar, then μ ≥ n − 5.^[1]^[5]

Graph minors

A 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:

If H is a minor of G then

\mu (H)\leq \mu (G)

.^[2]

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. Colin de Verdière (1990) lists these sets of forbidden minors 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 linklessly embeddable graphs as the graphs with μ ≤ 4 and as the graphs with no Petersen family minor.^[4] For k = 5 the set of forbidden minors includes the 78 graphs of the Heawood family, and it is conjectured that this list is complete.^[6]

Chromatic number

Colin de Verdière (1990) conjectured that any graph with Colin de Verdière invariant μ may be colored with at most μ + 1 colors. For instance, the linear forests have invariant 1, and can be 2-colored; the outerplanar graphs have invariant two, and can be 3-colored; the planar graphs have invariant 3, and (by the four color theorem) can be 4-colored.

For graphs with Colin de Verdière invariant at most four, the conjecture remains true; these are the linklessly embeddable graphs, and the fact that they have chromatic number at most five is a consequence of a proof by Neil Robertson, Paul Seymour, and Robin Thomas (1993) of the Hadwiger conjecture for K₆-minor-free graphs.

Other properties

If a graph has crossing number $k$ , it has Colin de Verdière invariant at most $k+3$ . For instance, the two Kuratowski graphs $K_{5}$ and $K_{3,3}$ can both be drawn with a single crossing, and have Colin de Verdière invariant at most four.^[2]

Influence

The Colin de Verdière invariant is defined through a class of matrices corresponding to the graph instead of just a single matrix. Along the same lines other graph parameters can be defined and studied, such as the minimum rank, minimum semidefinite rank and minimum skew rank.

Notes

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j van der Holst, Lovász & Schrijver (1999).
^ ^a ^b ^c ^d ^e ^f Colin de Verdière (1990).
^ 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 or claw minor.
^ ^a ^b Lovász & Schrijver (1998).
^ ^a ^b ^c Kotlov, Lovász & Vempala (1997).
^ Hein van der Holst (2006). "Graphs and obstructions in four dimensions" (PDF). Journal of Combinatorial Theory, Series B. 96 (3): 388–404. doi:10.1016/j.jctb.2005.09.004.

References

Colin de Verdière, Yves (1990), "Sur un nouvel invariant des graphes et un critère de planarité", Journal of Combinatorial Theory, Series B, 50 (1): 11–21, doi:10.1016/0095-8956(90)90093-F. Translated by Neil J. Calkin as Colin de Verdière, Yves (1993), "On a new graph invariant and a criterion for planarity", in Robertson, Neil; Seymour, Paul (eds.), Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors, Contemporary Mathematics, vol. 147, American Mathematical Society, pp. 137–147.
van der Holst, Hein; Lovász, László; Schrijver, Alexander (1999), "The Colin de Verdière graph parameter", Graph Theory and Combinatorial Biology (Balatonlelle, 1996), Bolyai Soc. Math. Stud., vol. 7, Budapest: János Bolyai Math. Soc., pp. 29–85, archived from the original on 2016-03-03, retrieved 2010-08-06.
Kotlov, Andrew; Lovász, László; Vempala, Santosh (1997), "The Colin de Verdiere number and sphere representations of a graph", Combinatorica, 17 (4): 483–521, doi:10.1007/BF01195002, archived from the original on 2016-03-03, retrieved 2010-08-06
Lovász, László; Schrijver, Alexander (1998), "A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs", Proceedings of the American Mathematical Society, 126 (5): 1275–1285, doi:10.1090/S0002-9939-98-04244-0.
Robertson, Neil; Seymour, Paul; Thomas, Robin (1993), "Hadwiger's conjecture for K₆-free graphs" (PDF), Combinatorica, 13 (3): 279–361, doi:10.1007/BF01202354, archived from the original (PDF) on 2009-04-10, retrieved 2010-08-06.