combin · Graph¶

29 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example¶

The complete checked program Canonically label a cycle and compute its automorphism group uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Canonically label a cycle and compute its automorphism group."""

from noether.combin import Graph


def main() -> None:
    cycle = Graph.from_unweighted(
        5,
        [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)],
        directed=False,
    )
    canonical = cycle.canonical_label()
    assert canonical.verify(cycle)
    assert canonical.automorphism_group.order == 10
    assert cycle.laplacian()[0] == [2, -1, 0, 0, -1]
    print(f"Aut(C5) has order {canonical.automorphism_group.order}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/21_graph_symmetry.py.

`combin.Graph`¶

The thin Python projection of Noether's CSR graph value.

Detailed contract

The thin Python projection of Noether's CSR graph value.

Canonical defining content is the wrapped graph_core::Graph value: (order, edges, directed). Serialization delegates to that value and never exposes its derived CSR layout. The optional display name is deliberately not mathematical identity.

Construct via Graph.from_edges (with weights) or Graph.from_unweighted (every edge weight = 1). Directed or undirected; parallel edges and self-loops are accepted verbatim (graph-core is multigraph-capable).

PythonRust

Graph(n_vertices: int, edges: Sequence[Sequence[int]], name: str | None=None)

noether::combin::Graph · Direct Rust declaration

pub struct Graph { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/21_graph_symmetry.py:5 · examples/21_graph_symmetry.py:9 · validation/unit/combin/test_acceptance_workload.py:10 · validation/unit/combin/test_acceptance_workload.py:90 · validation/unit/combin/test_graph_facade.py:110 · validation/unit/combin/test_graph_facade.py:114

`combin.Graph.eq`¶

Test mathematical equality with another value.

PythonRust

(self, other: object) -> bool

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

`combin.Graph.automorphism_group`¶

The exact automorphism group acting on this graph's vertex ids.

PythonRust

(self) -> PermutationGroup

noether::combin::Graph::automorphism_group · Direct Rust declaration

fn automorphism_group(&self) -> Result<PermutationGroup, GraphSymmetryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: examples/21_graph_symmetry.py:16 · examples/21_graph_symmetry.py:18 · examples/29_integral_line_configurations.py:10 · validation/unit/category/test_category.py:55 · validation/unit/combin/test_acceptance_workload.py:31 · validation/unit/combin/test_graph_facade.py:91

`combin.Graph.bfs`¶

Breadth-first search from source.

Detailed contract

Breadth-first search from source. Returns a [Traversal] (order / predecessor / depth / path_to). Raises IndexError if source >= order.

Run breadth-first search from source. The traversal uses the graph’s deterministic ascending-neighbor order. §Errors GraphError::VertexOutOfRange if source >= self.order().

PythonRust

(self, source: VertexId) -> GraphTraversal

noether::combin::Graph::bfs · Direct Rust declaration

pub fn bfs(&self, source: u32) -> Result<Traversal, GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:27 · validation/unit/combin/test_graph_facade.py:71 · validation/parity/u1_graph_core.py:274 · validation/parity/u1_graph_core.py:310 · validation/parity/u1_graph_core.py:41

`combin.Graph.canonical_label`¶

Compute an exact canonical labeling of this directed, weighted multigraph.

Detailed contract

Compute an exact canonical labeling of this directed, weighted multigraph. The result also carries the native automorphism group found by the same bounded individualization/refinement search.

PythonRust

(self) -> CanonicalLabel

noether::combin::Graph::canonical_label · Direct Rust declaration

fn canonical_label(&self) -> Result<CanonicalLabel, GraphSymmetryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: examples/21_graph_symmetry.py:14 · validation/unit/category/test_category.py:16 · validation/unit/category/test_category.py:17 · validation/unit/combin/test_graph_facade.py:84 · validation/unit/combin/test_graph_facade.py:85 · validation/unit/combinatorics/test_poset.py:106

`combin.Graph.connected_components`¶

Connected (weak, for digraphs) component labelling.

Detailed contract

Connected (weak, for digraphs) component labelling. Returns a [Components]. Never fails.

Compute the canonical weakly connected-component labeling.

PythonRust

(self) -> ConnectedComponents

noether::combin::Graph::connected_components · Direct Rust declaration

pub fn connected_components(&self) -> Components

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:32 · validation/unit/combinatorics/test_poset.py:82 · validation/parity/combinatorics/test_tree_core_parity.py:258 · validation/parity/u1_graph_core.py:310 · validation/parity/u1_graph_core.py:43 · validation/backends/graph/igraph_oracle.py:10

`combin.Graph.csr_adjacency`¶

The raw CSR (offsets, targets) arrays.

Detailed contract

The raw CSR (offsets, targets) arrays.

offsets has length order + 1; vertex v's neighbours are targets[offsets[v] : offsets[v + 1]], sorted ascending. Returns (list[int], list[int]).

PythonRust

(self) -> tuple[list[int], list[VertexId]]

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/parity/u1_graph_core.py:30 · validation/parity/u1_graph_core.py:309

`combin.Graph.degree`¶

The out-degree of v (or, for an undirected graph, the degree, with a self-loop contributing 1).

Detailed contract

The out-degree of v (or, for an undirected graph, the degree, with a self-loop contributing 1).

Raises IndexError if v >= order.

The out-degree of v (for an undirected graph, the degree, with a self-loop contributing 1). §Panics Panics if v >= order().

PythonRust

(self, vertex: VertexId) -> int

noether::combin::Graph::degree · Direct Rust declaration

pub fn degree(&self, v: u32) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:18 · examples/15_groebner_twisted_cubic.py:15 · examples/15_groebner_twisted_cubic.py:28 · examples/31_local_fields_and_precision.py:37 · examples/33_simplicial_cohomology_operations.py:26

`combin.Graph.degree_sequence`¶

The degree of each vertex, sorted descending (undirected; a self-loop would contribute 2, but simple graphs have none).

Detailed contract

The degree of each vertex, sorted descending (undirected; a self-loop would contribute 2, but simple graphs have none).

PythonRust

(self) -> list[int]

noether::combin::Graph::degree_sequence · Direct Rust declaration

fn degree_sequence(&self) -> Vec<i64>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:93 · validation/corpus/combin/bernhart-kainen-algebraic-graph.py:24 · validation/corpus/combin/bernhart-kainen-algebraic-graph.py:43 · validation/corpus/linalg/graph-minimum-rank-witness.py:27

`combin.Graph.dfs`¶

Depth-first search from source.

Detailed contract

Depth-first search from source. Iterative (no recursion). Returns a [Traversal]. Raises IndexError if source >= order.

Run iterative depth-first search from source. §Errors GraphError::VertexOutOfRange if source >= self.order().

PythonRust

(self, source: VertexId) -> GraphTraversal

noether::combin::Graph::dfs · Direct Rust declaration

pub fn dfs(&self, source: u32) -> Result<Traversal, GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: validation/parity/u1_graph_core.py:310 · validation/parity/u1_graph_core.py:41

`combin.Graph.edge_count`¶

The number of edges (each undirected pair counted once; parallel edges counted with multiplicity).

Detailed contract

The number of edges (each undirected pair counted once; parallel edges counted with multiplicity).

The number of edges as supplied (each undirected pair counted once; parallel edges counted with multiplicity).

PythonRust

(self) -> int

noether::combin::Graph::edge_count · Direct Rust declaration

pub fn edge_count(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:23 · validation/parity/u1_graph_core.py:34 · validation/parity/u1_graph_core.py:38 · validation/backends/graph/igraph_oracle.py:111 · validation/backends/graph/igraph_oracle.py:131 · validation/backends/graph/invariants.py:118

`combin.Graph.edges`¶

The defining edges as sorted [u, v] pairs, with weights dropped.

Detailed contract

The defining edges as sorted [u, v] pairs, with weights dropped. For an undirected simple graph this is the canonical edge multiset in construction order.

The user-supplied edges, in insertion order.

PythonRust

(self) -> list[list[int]]

noether::combin::Graph::edges · Direct Rust declaration

pub fn edges(&self) -> &[Edge]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:101 · validation/unit/infra/test_magma_release_protocol.py:256 · validation/unit/infra/test_magma_release_protocol.py:258 · validation/parity/combinatorics/test_crystals_parity.py:82 · validation/parity/combinatorics/test_crystals_parity.py:88 · validation/parity/combinatorics/test_tree_core_parity.py:119

`combin.Graph.from_edges`¶

Construct a graph from a list of (source, target, weight) triples.

Detailed contract

Construct a graph from a list of (source, target, weight) triples. directed is a required positional (pass False for an undirected graph); it has no default by design.

Raises ValueError if edges is not a list of weighted 3-tuples (for unweighted (source, target) pairs use from_unweighted), and IndexError if any endpoint is >= n.

Construct a graph from an edge list over n vertices. edges is (source, target, weight) triples. Pass weight 1 for an unweighted graph (or use [graph_from_unweighted]). directed selects the directed or undirected interpretation. Parallel edges and self-loops are accepted verbatim — this is a multigraph-capable constructor. §Errors GraphError::VertexOutOfRange if any endpoint is >= n. The empty graph (n = 0, no edges) and the edgeless graph (n > 0, no edges) are both valid.

PythonRust

(n: int, edges: Sequence[WeightedEdge], directed: bool) -> Graph

noether::combin::Graph::from_edges · Direct Rust declaration

pub fn from_edges( n: usize, edges: &[(u32, u32, i64)], directed: bool, ) -> Result<Graph, GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:114 · validation/unit/combin/test_graph_facade.py:40 · validation/unit/combin/test_graph_facade.py:75 · validation/parity/combinatorics/test_tree_pyo3.py:17 · validation/parity/combinatorics/test_tree_pyo3.py:197 · validation/parity/combinatorics/test_tree_pyo3.py:280

`combin.Graph.from_json`¶

Deserialize and revalidate a graph's defining content.

PythonRust

(value: str) -> Graph

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

`combin.Graph.from_unweighted`¶

Construct an unweighted graph from a list of (source, target) pairs.

Detailed contract

Construct an unweighted graph from a list of (source, target) pairs. Every edge gets weight 1.

Raises IndexError if any endpoint is >= n.

Construct an unweighted graph from (source, target) pairs. Every edge receives weight 1. Parallel edges and self-loops are accepted, as in Graph::from_edges. §Errors GraphError::VertexOutOfRange if any endpoint is >= n.

PythonRust

(n: int, edges: Sequence[tuple[VertexId, VertexId]], directed: bool) -> Graph

noether::combin::Graph::from_unweighted · Direct Rust declaration

pub fn from_unweighted( n: usize, edges: &[(u32, u32)], directed: bool, ) -> Result<Graph, GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/21_graph_symmetry.py:9 · validation/unit/combin/test_acceptance_workload.py:90 · validation/unit/combin/test_graph_facade.py:110 · validation/unit/combin/test_graph_facade.py:19 · validation/unit/combin/test_graph_facade.py:55 · validation/unit/combin/test_graph_facade.py:68

`combin.Graph.is_directed`¶

True if this is a directed graph, False for undirected.

Detailed contract

true if this is a directed graph.

PythonRust

(self) -> bool

noether::combin::Graph::is_directed · Direct Rust declaration

pub fn is_directed(&self) -> bool

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combinatorics/test_poset.py:25

`combin.Graph.laplacian`¶

The Laplacian matrix L = D - A as a dense list[list[int]], computed from the defining edges (treated as undirected).

Detailed contract

The Laplacian matrix L = D - A as a dense list[list[int]], computed from the defining edges (treated as undirected).

PythonRust

(self) -> list[list[int]]

noether::combin::Graph::laplacian · Direct Rust declaration

fn laplacian(&self) -> Vec<Vec<i64>>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/21_graph_symmetry.py:17 · validation/unit/combin/test_acceptance_workload.py:94

`combin.Graph.laplacian_spectrum`¶

The Laplacian spectrum: eigenvalues of [laplacian] in ascending order.

Detailed contract

The Laplacian spectrum: eigenvalues of [laplacian] in ascending order. The Laplacian is symmetric (undirected by construction, like laplacian() itself), so every eigenvalue is real; the smallest is always 0.0, with multiplicity equal to the number of connected components. Bound-tier (native f64 via LAPACK dsyev) — the Laplacian is exact-integer, so this trades exactness for the speed and familiarity of a plain numeric spectrum (the convention every other graph library uses); raises on the (pathological) case where LAPACK fails to converge.

PythonRust

(self) -> list[float]

noether::combin::Graph::laplacian_spectrum · Direct Rust declaration

fn laplacian_spectrum(&self) -> Result<Vec<f64>, GraphInvariantError>

Contract	Value
Guarantee	documented numeric bound
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:96

`combin.Graph.n_edges`¶

The number of edges (len(edges)).

PythonRust

(self) -> int

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:92

`combin.Graph.n_vertices`¶

The number of vertices (alias of order).

PythonRust

(self) -> int

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/parity/polyhedral/test_convex_hull_parity.py:190

`combin.Graph.name`¶

The optional human-readable name (e.g.

Detailed contract

The optional human-readable name (e.g. "K_4"), or None.

PythonRust

(self) -> str | None

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/12_number_field_ideals.py:12 · validation/unit/commalg/test_polynomial_facade.py:345 · validation/unit/commalg/test_polynomial_facade.py:44 · validation/unit/commalg/test_polynomial_facade.py:61 · validation/unit/commalg/test_polynomial_facade.py:65 · validation/unit/commalg/test_polynomial_facade.py:72

`combin.Graph.neighbors`¶

Vertex v's out-neighbours (directed) or all neighbours (undirected), in ascending target-id order, with multiplicity.

Detailed contract

Vertex v's out-neighbours (directed) or all neighbours (undirected), in ascending target-id order, with multiplicity.

Returns a list[int]. Raises IndexError if v >= order.

Vertex v’s out-neighbours (for a directed graph) or all neighbours (for an undirected graph), sorted ascending. Returns the slice with multiplicity — a vertex reached by k parallel edges appears k times. §Panics Panics if v >= order(). This is an internal index accessor, not a fallible entry point — callers within the crate hold v < n by construction. External callers should bound-check.

PythonRust

(self, vertex: VertexId) -> list[VertexId]

noether::combin::Graph::neighbors · Direct Rust declaration

pub fn neighbors(&self, v: u32) -> &[u32]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:24 · validation/unit/combin/test_graph_facade.py:51 · validation/unit/combin/test_graph_facade.py:73 · validation/corpus/combin/johnson-strongly-regular-graph.py:24 · validation/corpus/combin/johnson-strongly-regular-graph.py:25

`combin.Graph.order`¶

The number of vertices.

Detailed contract

The number of vertices. Valid vertex ids are 0..order.

The number of vertices. Valid vertex ids are 0..order().

PythonRust

(self) -> int

noether::combin::Graph::order · Direct Rust declaration

pub fn order(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/11_qadic_witt_vectors.py:14 · examples/12_number_field_ideals.py:13 · examples/12_number_field_ideals.py:14 · examples/12_number_field_ideals.py:17 · examples/15_groebner_twisted_cubic.py:18 · examples/19_permutation_groups.py:13

`combin.Graph.reduced_laplacian`¶

The reduced Laplacian: the Laplacian with the row and column of delete_vertex removed.

Detailed contract

The reduced Laplacian: the Laplacian with the row and column of delete_vertex removed. Its cokernel is the sandpile group K(G). Raises IndexError if delete_vertex is out of range.

PythonRust

(self, delete_vertex: VertexId=0) -> list[list[int]]

noether::combin::Graph::reduced_laplacian · Direct Rust declaration

fn reduced_laplacian( &self, delete_vertex: usize, ) -> Result<Vec<Vec<i64>>, GraphInvariantError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:95 · validation/corpus/combin/igraph-grid-spanning-trees.py:14 · validation/corpus/combin/igraph-k4-spanning-trees.py:15

`combin.Graph.shortest_path`¶

A shortest s–t path.

Detailed contract

A shortest s–t path. Uses BFS when every edge weight is 1 and Dijkstra otherwise.

Returns a [ShortestPath] when t is reachable, None when it is not. Raises IndexError if s / t >= order or ValueError on a negative weight reached by Dijkstra.

Compute a shortest path from source to target. Unit-weight graphs use BFS and other nonnegative graphs use Dijkstra. A missing path is Ok(None). §Errors GraphError::VertexOutOfRange if either endpoint is outside the graph, or GraphError::NegativeWeight if Dijkstra encounters a negative edge weight.

PythonRust

(self, source: VertexId, target: VertexId) -> ShortestPath | None

noether::combin::Graph::shortest_path · Direct Rust declaration

pub fn shortest_path( &self, source: u32, target: u32, ) -> Result<Option<ShortestPath>, GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	released
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:46 · validation/unit/combin/test_graph_facade.py:77 · validation/parity/u1_graph_core.py:311 · validation/parity/u1_graph_core.py:44 · validation/corpus/combin/cozero-divisor-wiener-index.py:21

`combin.Graph.subgraph`¶

The induced subgraph on vertices.

Detailed contract

The induced subgraph on vertices.

Returns a 2-tuple, not a bare Graph (the name notwithstanding): (sub: Graph, vertex_map: list[int]), where new vertex i is the original vertex vertex_map[i] — you need vertex_map to map results back to the parent's vertex ids. Unpack it as sub, vmap = g.subgraph(...). Raises IndexError if any entry of vertices is out of range.

The induced subgraph on vertices. Returns a new graph whose vertices are exactly the distinct entries of vertices, renumbered 0..k in the order they first appear, together with every edge of self both of whose endpoints lie in the set. Directedness, weights, and edge multiplicity are preserved. The returned Graph is paired with the vertex map — a Vec whose i-th entry is the original id of new vertex i — so labels can be carried across (see Labeling::subgraph). §Errors GraphError::VertexOutOfRange if any entry of vertices is >= order().

PythonRust

(self, vertices: Sequence[VertexId]) -> tuple[Graph, list[VertexId]]

noether::combin::Graph::subgraph · Direct Rust declaration

pub fn subgraph( &self, vertices: &[u32], ) -> Result<(Graph, Vec<u32>), GraphError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:59 · validation/unit/combin/test_graph_facade.py:60 · validation/parity/u1_graph_core.py:106 · validation/parity/u1_graph_core.py:110 · validation/parity/u1_graph_core.py:286 · validation/parity/u1_graph_core.py:287

`combin.Graph.to_json`¶

Serialize the graph's validated defining content.

PythonRust

(self) -> str

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

`combin.Graph.transpose`¶

The reverse digraph.

Detailed contract

The reverse digraph. The identity on undirected graphs (returns a clone).

The reverse digraph: every edge u → v becomes v → u. For an undirected graph transposition is the identity — the returned graph equals self. For a directed graph the vertex set, weights, and edge multiplicity are preserved; only edge orientation flips. transpose(transpose(g)) == g for every graph g (the involution invariant — see the crate tests).

PythonRust

(self) -> Graph

noether::combin::Graph::transpose · Direct Rust declaration

pub fn transpose(&self) -> Graph

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: examples/39_kahler_differentials.py:26 · examples/distributed/f2_linear_algebra.py:31 · validation/unit/combin/test_graph_facade.py:51 · validation/unit/commalg/test_power_series_facade.py:347 · validation/unit/commalg/test_quotient_facade.py:162 · validation/unit/infra/test_boundary_properties.py:54

`combin.Graph.weighted_edges`¶

The defining (source, target, weight) triples in construction order.

Detailed contract

The defining (source, target, weight) triples in construction order. This is the lossless edge view for directed, weighted, and multigraph inputs.

PythonRust

(self) -> list[WeightedEdge]

noether::combin::Graph · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, graph-core, perm-core
GIL	attached / bounded
Semantic ID	noether/combin/Graph@1

Executable usage: validation/unit/combin/test_graph_facade.py:50 · validation/unit/combinatorics/test_poset.py:26 · validation/parity/combinatorics/test_poset_parity.py:73

combin · Graph¶

Worked example¶

combin.Graph¶

combin.Graph.__eq__¶

combin.Graph.automorphism_group¶

combin.Graph.bfs¶

combin.Graph.canonical_label¶

combin.Graph.connected_components¶

combin.Graph.csr_adjacency¶

combin.Graph.degree¶

combin.Graph.degree_sequence¶

combin.Graph.dfs¶

combin.Graph.edge_count¶

combin.Graph.edges¶

combin.Graph.from_edges¶

combin.Graph.from_json¶

combin.Graph.from_unweighted¶

combin.Graph.is_directed¶

combin.Graph.laplacian¶

combin.Graph.laplacian_spectrum¶

combin.Graph.n_edges¶

combin.Graph.n_vertices¶

combin.Graph.name¶

combin.Graph.neighbors¶

combin.Graph.order¶

combin.Graph.reduced_laplacian¶

combin.Graph.shortest_path¶

combin.Graph.subgraph¶

combin.Graph.to_json¶

combin.Graph.transpose¶

combin.Graph.weighted_edges¶

`combin.Graph`¶

`combin.Graph.eq`¶

`combin.Graph.automorphism_group`¶

`combin.Graph.bfs`¶

`combin.Graph.canonical_label`¶

`combin.Graph.connected_components`¶

`combin.Graph.csr_adjacency`¶

`combin.Graph.degree`¶

`combin.Graph.degree_sequence`¶

`combin.Graph.dfs`¶

`combin.Graph.edge_count`¶

`combin.Graph.edges`¶

`combin.Graph.from_edges`¶

`combin.Graph.from_json`¶

`combin.Graph.from_unweighted`¶

`combin.Graph.is_directed`¶

`combin.Graph.laplacian`¶

`combin.Graph.laplacian_spectrum`¶

`combin.Graph.n_edges`¶

`combin.Graph.n_vertices`¶

`combin.Graph.name`¶

`combin.Graph.neighbors`¶

`combin.Graph.order`¶

`combin.Graph.reduced_laplacian`¶

`combin.Graph.shortest_path`¶

`combin.Graph.subgraph`¶

`combin.Graph.to_json`¶

`combin.Graph.transpose`¶

`combin.Graph.weighted_edges`¶