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.
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.
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. 2. 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) — more general patterns this builds on
-
Colin de Verdière graph invariant is a kind of Invariance Prime
The proposed strict upward parent is
prime:invariance.
Hierarchy path (1) — routes to 1 parentless root
- Colin de Verdière graph invariant → Invariance
Neighborhood in Abstraction Space¶
Colin de Verdière graph invariant sits in a crowded region of the domain-specific corpus (6th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Invariants & Constructions (49 abstractions)
Nearest neighbors
- Integral graph — 0.96
- Strong product of graphs — 0.93
- Conductance (graph theory) — 0.93
- Dually chordal graph — 0.92
- Split graph — 0.92
Computed from structural-signature embeddings · 2026-09-08