Integral graph

A finite graph whose adjacency matrix has only integer eigenvalues.

Version

v2 · 2026-09-19 · History

Domain-specific #

5058

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Spectral Graph Theory → Mathematics

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.

How would you explain it like I'm…

 

No faithful explanation at this level. The property concerns the eigenvalues of the adjacency matrix. Any picture simple enough for a five-year-old collapses into 'a graph labeled with whole numbers' or 'whole numbers of dots and lines', which is true of every graph and is not what integral means.

Whole-Number Fingerprint Graphs

A graph here is a bunch of dots with lines connecting some pairs. You can write it as a table of 0s and 1s, with a 1 wherever two dots are connected. From that table, mathematicians can compute a special list of 'fingerprint numbers' for the graph. If every number on that list is a whole number, with no messy decimals, the graph is called an integral graph. Just counting dots or lines doesn't tell you; you have to work out the fingerprint numbers.

Integer-Spectrum Graph

An integral graph is a graph whose adjacency matrix has only integer eigenvalues. The adjacency matrix has a 1 in row i, column j when vertices i and j are connected, and 0 otherwise. You form its characteristic polynomial and check whether every root, counted with multiplicity, is an integer. Relabeling the vertices doesn't change the answer, so it is a property of the graph itself. Having integer coefficients in the polynomial isn't enough, since the roots can still be irrational. Also, "integral" here means adjacency eigenvalues, not the eigenvalues of the Laplacian matrix, which is a different notion.

 

An integral graph is one whose adjacency spectrum consists entirely of integers. Fix the graph class (typically simple, undirected) and form the adjacency matrix A under some vertex ordering. Compute the characteristic polynomial det(xI − A), and check that every root in the eigenvalue multiset, multiplicities included, is an integer. Since A is an integer matrix, the polynomial has integer coefficients, but that alone does not ensure integral roots. The property is invariant under graph isomorphism, because reordering vertices conjugates A by a permutation matrix and preserves the spectrum. The convention matters. Unless qualified, 'integral' refers to the adjacency matrix, not the Laplacian; 'Laplacian-integral' is a distinct notion. Graphs with merely rational spectra form a separate category. Allowing loops, multiple edges, or directed edges changes the matrix and hence the spectrum.

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

1. 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) — outgoing typed relations

• Integral graph is a kind of Classification Prime

  It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

• Integral graph → is a kind of Classification

Neighborhood in Abstraction Space

Integral graph sits in a crowded region of the domain-specific corpus (2^nd 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

• 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 · 2026-10-25