Cyclomatic number¶
The dimension of an undirected graph's cycle space, equal to the minimum number of edges whose removal makes it acyclic and computed as edges minus vertices plus connected components.
Core Idea¶
Also called circuit rank or cycle rank, the number counts independent cycles, is additive over components and equals the number of non-tree edges relative to any spanning forest. A spanning forest contributes one fewer edge than vertices in each component; every remaining edge closes one independent fundamental cycle, yielding e-v+c basis elements. 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¶
Cyclomatic number belongs to graph theory and is useful where the analyst can specify the typed graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite undirected graph, loop and parallel-edge convention, vertex and edge counts, connected components, spanning forest, cycle-space coefficient field, independent-cycle basis, edge-removal interpretation, and formula are explicit. The scope is broad within that domain but bounded by the need for the finite undirected graph, loop and parallel-edge convention, vertex and edge counts, connected components, spanning forest, cycle-space coefficient field, independent-cycle basis, edge-removal interpretation, and formula are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite undirected graph, loop and parallel-edge convention, vertex and edge counts, connected components, spanning forest, cycle-space coefficient field, independent-cycle basis, edge-removal interpretation, and formula 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. A bare label is insufficient because the name Cyclomatic number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
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 Cyclomatic number. Cyclomatic number 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 graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite undirected graph, loop and parallel-edge convention, vertex and edge counts, connected components, spanning forest, cycle-space coefficient field, independent-cycle basis, edge-removal interpretation, and formula are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph theory because they reuse the typed graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, A spanning forest contributes one fewer edge than vertices in each component; every remaining edge closes one independent fundamental cycle, yielding e-v+c basis elements., and type the carrier, state every parameter and convention in the definition, test that the finite undirected graph, loop and parallel-edge convention, vertex and edge counts, connected components, spanning forest, cycle-space coefficient field, independent-cycle basis, edge-removal interpretation, and formula are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Cyclomatic number Domain-specific
Parents (1) — more general patterns this builds on
-
Cyclomatic number is a kind of Dimension Prime
The proposed strict upward parent is
prime:dimension.
Hierarchy path (1) — routes to 1 parentless root
- Cyclomatic number → Dimension
Neighborhood in Abstraction Space¶
Cyclomatic number sits in a crowded region of the domain-specific corpus (6th 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
- Pancyclic graph — 0.94
- Starlike tree — 0.93
- Split graph — 0.93
- Treewidth — 0.93
- Crossing number (graph theory) — 0.93
Computed from structural-signature embeddings · 2026-09-08