DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Binary adjacency matrix of an ordered finite simple graph

Definition

Let $G = (V, E)$ be a finite simple graph with a fixed order $V = (v_{0}, \dots, v_{n - 1})$ . Put $F_{2} : = Z / 2 Z$ ; the field law follows from For every prime $p$ , the two operations on $Z / p$ make it a field at $p = 2$ , and this field has order two by Finite fields and their order. Define the binary adjacency matrix $A_{2} (G) \in M_{n} (F_{2})$ by

$(A_{2} (G))_{j k} = {\begin{cases} 1, & {v_{j}, v_{k}} \in E, \\ 0, & {v_{j}, v_{k}} \notin E . \end{cases}$

The indices use the ordered bit coordinates of Big-endian encoding of ordered bit strings. Since edges are unordered pairs of distinct vertices, $A_{2} (G)$ is symmetric and has zero diagonal. This is a matrix over $F_{2}$ , using Finite rectangular matrices over a commutative ring, their entries, rows and columns; it is not the root library's real-valued adjacency matrix. When $V = \emptyset$ , $A_{2} (G)$ is the unique $0 \times 0$ matrix.

Depends on

Used by

Dependency tree · two levels

24 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

Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Measurement-based quantum computation on cluster states, arXiv:quant-ph/0301052, p. 4 (standard reference, not scraped)
M. Hein et al., Entanglement in graph states and its applications, arXiv:quant-ph/0602096, §2.2.4, p. 23 (standard reference, not scraped)