Colin de Verdière graph invariant¶

A minor-monotone graph parameter defined by the maximum corank of a constrained symmetric matrix with one negative eigenvalue and the Strong Arnold property.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3739
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Spectral Graph Theory → Mathematics
Aliases: Colin de Verdière invariant

Core Idea¶

The invariant couples a graph's off-diagonal sign pattern to matrix inertia and a transversality condition, with small threshold values characterizing paths, outerplanar, planar and linklessly embeddable graph families. Admissible matrices encode adjacency through negative off-diagonal entries, spectral and Strong Arnold constraints exclude degeneracy and maximizing nullity yields a graph invariant stable under minors. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

No faithful explanation at this level. All three generators agree that a child-level picture turns the invariant into 'a score for how hard a dot-and-line drawing is to draw flat', collapsing a maximum-nullity invariant of sign-constrained matrices into a drawing measure, when the drawing families only characterize its small values.

The Matrix Score of a Graph

Take a network of dots joined by lines. Build a square table of numbers for it: where two dots are joined you must put a negative number, where they aren't joined you put zero, and the diagonal can be anything. The table also has to pass two special tests that stop it from being a lucky, fragile accident. Among all tables that pass, the Colin de Verdière invariant is the biggest number of independent 'ways' the table can turn an input into all zeros. The surprise is that small values of this number match networks you can draw in simple ways, such as flat on paper without crossings.

Matrix Nullity Graph Invariant

The Colin de Verdière invariant is a number attached to a finite simple graph using matrices. You consider symmetric matrices whose off-diagonal entries are negative exactly where two vertices are joined by an edge and zero where they are not, with the diagonal free. Such a matrix must have exactly one negative eigenvalue and must satisfy a nondegeneracy condition called the Strong Arnold Property. The invariant is the largest nullity, meaning the dimension of the space of vectors the matrix sends to zero, over all such admissible matrices. It does not increase when you take minors of the graph, meaning deleting or contracting edges. Its small values give exact descriptions of well-known graph families: paths, outerplanar graphs, planar graphs and linklessly embeddable graphs.

The Colin de Verdière invariant μ(G) of a finite simple graph G is defined spectrally. An admissible matrix M is real symmetric with M_ij < 0 when vertices i and j are adjacent and M_ij = 0 when distinct vertices are nonadjacent, with diagonal entries unconstrained. Admissibility further requires that M have exactly one negative eigenvalue and satisfy the Strong Arnold Property, a transversality condition that excludes degenerate matrices whose corank could be perturbed away. μ(G) is the maximum corank, or nullity, over all admissible matrices. The invariant is minor-monotone, so it is stable under taking graph minors. Small threshold values of μ characterize, in turn, paths, outerplanar graphs, planar graphs and linklessly embeddable graphs, tying a spectral quantity to embeddability. The notion is specific to this sign-pattern, inertia and transversality package, not spectral graph theory in general.

Scope of Application¶

Colin de Verdière graph invariant belongs to spectral graph theory and is useful where the analyst can specify the typed spectral graph theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization are explicit. The scope is broad within that domain but bounded by the need for the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Colin de Verdière graph invariant can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Colin de Verdière graph invariant. Colin de Verdière graph invariant compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed spectral graph theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of spectral graph theory because they reuse the typed spectral graph theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Admissible matrices encode adjacency through negative off-diagonal entries, spectral and Strong Arnold constraints exclude degeneracy and maximizing nullity yields a graph invariant stable under minors., and type the carrier, state every parameter and convention in the definition, test that the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Colin de Verdière graph invariant Domain-specific

Parents (1) — outgoing typed relations

Colin de Verdière graph invariant is a kind of Invariance Prime

It remains its own entry because its identity is fixed by the finite simple graph, admissible symmetric matrix sign pattern, exactly-one-negative-eigenvalue condition, Strong Arnold property, corank maximization and any minor or embedding characterization.

Hierarchy path (1) — routes to 1 parentless root

Colin de Verdière graph invariant → is a kind of Invariance

Neighborhood in Abstraction Space¶

Colin de Verdière graph invariant sits in a crowded region of the domain-specific corpus (4^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Coloring & Invariants (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25