Integral graph¶
A finite graph whose adjacency matrix has only integer eigenvalues.
Core Idea¶
The defining matrix convention must be adjacency rather than Laplacian unless qualified, graph simplicity and directionality affect the spectrum and integral characteristic-polynomial coefficients alone do not ensure integral roots. The graph is encoded by its adjacency matrix, the characteristic polynomial is formed and every spectral root is tested for membership in the integers. 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¶
Integral graph belongs to spectral graph theory and is useful where the analyst can specify the typed spectral graph theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the graph class and adjacency convention, vertex ordering and adjacency matrix, characteristic polynomial, multiset of eigenvalues including multiplicities, integrality test, invariance under graph isomorphism and distinctions from Laplacian-integral and rational-spectrum graphs are explicit. The scope is broad within that domain but bounded by the need for the graph class and adjacency convention, vertex ordering and adjacency matrix, characteristic polynomial, multiset of eigenvalues including multiplicities, integrality test, invariance under graph isomorphism and distinctions from Laplacian-integral and rational-spectrum graphs are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the graph class and adjacency convention, vertex ordering and adjacency matrix, characteristic polynomial, multiset of eigenvalues including multiplicities, integrality test, invariance under graph isomorphism and distinctions from Laplacian-integral and rational-spectrum graphs 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.
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 Integral graph. Integral graph 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the graph class and adjacency convention, vertex ordering and adjacency matrix, characteristic polynomial, multiset of eigenvalues including multiplicities, integrality test, invariance under graph isomorphism and distinctions from Laplacian-integral and rational-spectrum graphs 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The graph is encoded by its adjacency matrix, the characteristic polynomial is formed and every spectral root is tested for membership in the integers., and type the carrier, state every parameter and convention in the definition, test that the graph class and adjacency convention, vertex ordering and adjacency matrix, characteristic polynomial, multiset of eigenvalues including multiplicities, integrality test, invariance under graph isomorphism and distinctions from Laplacian-integral and rational-spectrum graphs are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Integral graph Domain-specific
Parents (1) — more general patterns this builds on
-
Integral graph is a kind of Classification Prime
The proposed strict upward parent is
prime:classification.
Hierarchy path (1) — routes to 1 parentless root
- Integral graph → Classification
Neighborhood in Abstraction Space¶
Integral graph sits in a crowded region of the domain-specific corpus (4th 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
- Colin de Verdière graph invariant — 0.96
- Chromatic polynomial — 0.93
- Split graph — 0.93
- Strong product of graphs — 0.93
- Join (graph theory) — 0.93
Computed from structural-signature embeddings · 2026-09-08