Colin de Verdière graph invariant

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

• (M1) for all ${\displaystyle i,j}$ with ${\displaystyle i\neq j}$: ${\displaystyle M_{i,j}<0}$ if ${\displaystyle \{i,j\}\in E}$, and ${\displaystyle M_{i,j}=0}$ if ${\displaystyle \{i,j\}\notin E}$;
• (M2) ${\displaystyle M}$ has exactly one negative eigenvalue, of multiplicity 1;
• (M3) there is no nonzero matrix ${\displaystyle X=(X_{i,j})\in \mathbb {R} ^{(n)}}$ such that ${\displaystyle MX=0}$ and such that ${\displaystyle X_{i,j}=0}$ if either ${\displaystyle i=j}$ or ${\displaystyle M_{i,j}\neq 0}$ 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 ${\displaystyle \mu (H)\leq \mu (G)}$.^[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 ${\displaystyle k}$, it has Colin de Verdière invariant at most ${\displaystyle k+3}$. For instance, the two Kuratowski graphs ${\displaystyle K_{5}}$ and ${\displaystyle K_{3,3}}$ 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

1. van der Holst, Lovász & Schrijver (1999).
2. Colin de Verdière (1990).
3. 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.
4. Lovász & Schrijver (1998).
5. Kotlov, Lovász & Vempala (1997).
6. 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.