Graph Theory Ontology

FLOW ALGEBRA
TITLE: Graph Theory Ontology
DOMAIN: Discrete Mathematics / Graph Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/graph-theory/graph-theory-ontology.html
VERSION: 1.0
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Formal mathematical model; live source page not independently verified.

1. PURPOSE
Represent graph-theoretic structures, operations, constraints, and derivations in a typed, machine-consumable ontology. Separate mathematically proved facts from heuristic claims.

2. TYPES
Vertex := opaque identifier
Edge := (Vertex, Vertex) for simple undirected graphs, unordered by interpretation
Arc := (Vertex, Vertex) for directed graphs, ordered
Graph := (V: Set[Vertex], E: Set[Edge])
Digraph := (V: Set[Vertex], A: Set[Arc])
WeightedGraph := (Graph, weight: E -> Real)
Walk := finite sequence of incident vertices and edges
Trail := walk with no repeated edge
Path := walk with no repeated vertex (except a closed path convention when explicitly stated)
Cycle := closed walk with no repeated internal vertex, length >= 3 in simple undirected graphs
Tree := connected acyclic undirected graph
Forest := acyclic undirected graph

3. CONSTRAINTS
For G=(V,E), every edge has endpoints in V.
Simple graphs exclude loops and parallel edges.
Directed graphs distinguish (u,v) from (v,u).
A weighted graph requires an explicit weight domain and any algorithmic preconditions.
A graph may be empty; algorithms must define behavior for empty or disconnected graphs.

4. PRIMITIVE RELATIONS
incident(v,e,G) iff v is an endpoint of e.
adjacent(u,v,G) iff {u,v} belongs to E (simple undirected case).
reachable(u,v,G) iff a path from u to v exists.
connected(G) iff every pair of vertices is mutually reachable (undirected).
subgraph(H,G) iff V(H) subseteq V(G) and E(H) subseteq E(G).
isomorphic(G,H) iff a vertex bijection preserves and reflects adjacency.

5. OPERATIONS
order(G) := |V|.
size(G) := |E|.
degree(v,G) := number of incident edges for simple undirected G.
neighbors(v,G) := {u in V : adjacent(u,v,G)}.
induced_subgraph(G,S) := (S, {e in E : both endpoints in S}), S subseteq V.
edge_delete(G,F) := (V, E\F), F subseteq E.
vertex_delete(G,S) := induced_subgraph(G,V\S).
complement(G) := (V, {{u,v}: u!=v and {u,v} notin E}), for simple undirected G.
transpose(D) := (V,{(v,u):(u,v) in A}), for directed D.

6. ALGEBRAIC LAWS AND INVARIANTS
Handshake lemma: sum_{v in V} degree(v,G) = 2|E| (finite undirected graph).
Number of odd-degree vertices is even (finite undirected graph).
A finite undirected tree with n>=1 vertices has n-1 edges.
For finite simple undirected G with n vertices, |E(complement(G))| = n(n-1)/2 - |E(G)|.
Graph isomorphism preserves order, size, degree multiset, and connectivity.
Every connected finite undirected graph has a spanning tree.

7. INFERENCE RULES
EDGE_ADJACENCY: {u,v} in E => adjacent(u,v,G).
UNDIRECTED_SYMMETRY: adjacent(u,v,G) => adjacent(v,u,G).
PATH_REACHABILITY: path(u,v,G) => reachable(u,v,G).
REACHABILITY_TRANSITIVITY: reachable(u,v,G) and reachable(v,w,G) => reachable(u,w,G).
TREE_CHARACTERISATION: finite undirected connected(G) and acyclic(G) => tree(G).
CYCLE_OBSTRUCTION: cycle(G) => not tree(G).
ISOMORPHISM_TRANSFER: isomorphic(G,H) and invariant_property(G) => invariant_property(H).

8. ALGORITHMIC FLOWS
BFS(G,s): visit vertices reachable from s by increasing unweighted path length; O(|V|+|E|) adjacency-list representation.
DFS(G,s): explore reachable vertices depth-first; O(|V|+|E|) adjacency-list representation.
DIJKSTRA(G,s): shortest paths from s with nonnegative edge weights; reject negative-weight inputs.
TOPOLOGICAL_SORT(D): return linear order respecting every arc iff D is acyclic; otherwise report cycle.
CONNECTED_COMPONENTS(G): partition vertices by undirected reachability.

9. SEMANTIC RELATIONSHIPS
GraphTheoryOntology -> specialises -> DiscreteMathematicsOntology
Graph -> has_component -> Vertex
Graph -> has_component -> Edge
Digraph -> has_component -> Arc
Tree -> subclass_of -> ConnectedAcyclicGraph
Path -> witnesses -> Reachability
GraphIsomorphism -> preserves -> GraphInvariant
Adjacency -> represented_by -> AdjacencyMatrix

10. SEMANTIC_EQUIVALENCE
For simple undirected graphs, {u,v} and {v,u} denote the same edge.
An adjacency list and adjacency matrix are equivalent representations only when they encode the same vertex set and adjacency relation.
Graph isomorphism is structural equivalence, not literal identity of vertex identifiers.
A directed arc is not semantically interchangeable with an undirected edge.

11. VALIDATION
CHECK endpoints_exist for every edge/arc.
CHECK no_loops_or_duplicates when graph_kind=simple.
CHECK degree_sum = 2*edge_count for finite undirected graphs.
CHECK tree => connected and acyclic.
CHECK algorithm_preconditions before execution.
CHECK witness paths for claimed reachability and cycles for claimed cyclicity.
CHECK negative claims with exhaustive or certified algorithms where required.

12. CONFIDENCE
formal_theorem: 1.0 when assumptions are satisfied and derivation is checked.
computed_result: 1.0 only when algorithm, inputs, and execution trace are verified.
heuristic_result: attach calibrated confidence and method; do not label as proof.
source_alignment: unverified until the published page is inspected.

13. FLOW_HASH
canonicalise: UTF-8, stable key ordering, explicit graph_kind, sorted vertex IDs and edge encodings.
method: SHA-256(canonical_flow_representation).
value: compute at publication; do not invent a digest.

14. UPDATE_MODEL
on_change(graph_vertices_or_edges): invalidate affected degree, adjacency, path, component and invariant caches.
on_change(ontology_rules): increment version; rerun validation and dependency checks.
provenance: record editor, timestamp, revision, and verified mathematical sources.

15. AI_CONSUMPTION
input: typed graph objects, graph_kind, constraints, query, provenance.
process: parse -> typecheck -> validate -> choose sound inference or algorithm -> verify -> emit.
output: conclusion, assumptions, witness/certificate when available, confidence, version, flow_hash.
safety: do not confuse reachability with adjacency; do not assume connectedness, simplicity, finiteness, or nonnegative weights without evidence.

END FLOW ALGEBRA