PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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The adjacency spectrum is an isomorphism invariant

Statement

If finite simple graphs $G$ and $H$ are isomorphic, then they have the same adjacency spectrum. In particular, cospectrality is an isomorphism invariant.

Facts & Assumptions

Given: Finite simple graphs $G$ and $H$ and an isomorphism $φ : G \to H$ .

[F1]

A graph isomorphism is a bijection on vertices that preserves and reflects adjacency (Graph isomorphisms, automorphisms and graph complements).

[F2]

The adjacency matrix records adjacency in the chosen vertex order (The adjacency matrix of a finite simple graph).

[F3]

The adjacency spectrum is the multiset of roots of the adjacency characteristic polynomial, listed in weakly decreasing order (Adjacency spectrum, spectral radius, and cospectral graphs, For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix).

Proof

technique · direct

1.1

Order the vertices of $G$ as $v_{1}, \dots, v_{n}$ and the vertices of $H$ as $φ (v_{1}), \dots, φ (v_{n})$ . In these orders the adjacency matrices $A (G)$ and $A (H)$ have the same entries, because [F1] and [F2] say that the $(i, j)$ entry is $1$ in either matrix exactly when $v_{i}$ and $v_{j}$ are adjacent in $G$ .

F1F2choose

2.1

Since the two matrices are equal after a relabelling of the basis, they have the same characteristic polynomial and therefore the same spectrum by [F3]. This is exactly the claimed invariance.

step 1.1F3∎

Depends on

Used by

Two cospectral graphs need not be isomorphic Counterexample

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

O. Pikhurko, Algebraic Methods in Combinatorics, Chapter 14 (standard reference, not scraped)