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 no edges;^[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 include 78 graphs of Heawood family, and it is conjectured that there are no more.^[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

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: 279–361, doi:10.1007/BF01202354, archived from the original (PDF) on 2009-04-10, retrieved 2010-08-06.