# 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 loopless 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) 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:

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 along the same lines, such as the minimum rank, minimum semidefinite rank and minimum skew rank.

## Notes

1. ^ 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.
2. ^ a b
3. ^ a b c
4. ^ 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.