Complete Graph Ontology

FLOW ALGEBRA
TITLE: Complete Graph Ontology
DOMAIN: Discrete Mathematics / Graph Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/graph-theory/complete-graph-ontology.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical synthesis; live page not independently verified.

1. ONTOLOGY
ontology CompleteGraphOntology extends GraphTheoryOntology
  entity Graph: G = (V, E)
  entity Vertex: v in V
  entity Edge: {u,v} in E
  entity CompleteGraph: K_n, n >= 0
  entity Order: n = |V|
  entity Size: m = |E|
  entity Degree: deg(v)
  entity Clique: S subseteq V
  entity ComplementGraph: complement(G)
  entity AdjacencyMatrix: A(G)
  entity ChromaticNumber: chi(G)
  entity CliqueNumber: omega(G)
  entity IndependenceNumber: alpha(G)

2. DEFINITIONS AND TYPE CONSTRAINTS
  simple_undirected(G) := undirected(G) AND loop_free(G) AND no_parallel_edges(G)
  complete(G) := simple_undirected(G) AND
    forall u,v in V: (u != v => {u,v} in E)
  K_n := unique complete simple undirected graph of order n, up to isomorphism
  empty_vertex_case: K_0 has no vertices and no edges
  singleton_case: K_1 has one vertex and no edges
  edge_case: K_2 has two vertices and one edge
  clique(S,G) := forall distinct u,v in S: {u,v} in E(G)
  maximal_clique(S,G) := clique(S,G) AND no proper superset of S is a clique
  maximum_clique(S,G) := clique(S,G) AND |S| = omega(G)

3. OPERATIONS
  construct_complete(V) -> (V, {{u,v}: u,v in V AND u != v})
  add_vertex(K_n) -> K_(n+1) by joining new vertex to every old vertex
  delete_vertex(K_n,v) -> K_(n-1) for n >= 1
  induced_subgraph(K_n,S) -> K_|S|
  complement(K_n) -> edgeless_graph(n)
  complement(edgeless_graph(n)) -> K_n
  union_edge_sets(G,H) -> graph(V, E(G) union E(H)) when vertex sets match
  adjacency(K_n) -> J_n - I_n

4. ALGEBRAIC LAWS AND INVARIANTS
  |V(K_n)| = n
  |E(K_n)| = n(n-1)/2
  forall v in V(K_n): deg(v) = n-1
  sum(deg(v) for v in V) = n(n-1) = 2|E|
  chi(K_n) = n for n >= 1; chi(K_0) = 0
  omega(K_n) = n for n >= 1; omega(K_0) = 0
  alpha(K_n) = 1 for n >= 1; alpha(K_0) = 0
  connected(K_n) for n >= 1
  diameter(K_n) = 1 for n >= 2; diameter(K_1) = 0
  number_of_triangles(K_n) = C(n,3)
  number_of_k_cliques(K_n) = C(n,k) for 0 <= k <= n
  number_of_spanning_trees(K_n) = n^(n-2) for n >= 2
  edge_connectivity(K_n) = n-1 for n >= 2
  vertex_connectivity(K_n) = n-1 for n >= 2
  regular(K_n, n-1) for n >= 1
  planar(K_n) iff n <= 4
  bipartite(K_n) iff n <= 2
  Hamiltonian(K_n) for n >= 3
  Eulerian(K_n) iff n is odd for n >= 1

5. SEMANTIC RELATIONSHIPS
  CompleteGraph IS_A SimpleUndirectedGraph
  CompleteGraph HAS_VERTEX Vertex
  CompleteGraph HAS_EDGE Edge
  CompleteGraph SATISFIES PairwiseAdjacency
  CompleteGraph IS RegularGraph
  CompleteGraph CONTAINS Clique
  CompleteGraph COMPLEMENT_OF EdgelessGraph
  CompleteGraph RELATED_TO GraphColouring
  CompleteGraph RELATED_TO ExtremalGraphTheory
  CompleteGraph RELATED_TO NetworkTopology

6. INFERENCE RULES
  IF simple_undirected(G) AND forall distinct u,v in V: adjacent(u,v)
    THEN complete(G)
  IF complete(G) AND |V| = n THEN |E| = n(n-1)/2
  IF complete(G) AND n >= 1 THEN forall v: deg(v) = n-1
  IF simple_undirected(G) AND |E| = n(n-1)/2 THEN complete(G)
  IF complete(G) AND S subseteq V THEN complete(induced_subgraph(G,S))
  IF complete(G) AND n >= 3 THEN contains_triangle(G)
  IF complete(G) AND n >= 5 THEN not planar(G)
  IF complete(G) AND n >= 3 THEN not bipartite(G)
  CAUTION: edge count alone does not establish completeness for multigraphs or graphs with loops.

7. VALIDATION
  validate graph_is_simple_and_undirected
  validate vertices_are_unique
  validate edges_have_distinct_endpoints_in_V
  validate no_duplicate_undirected_edges
  validate every_distinct_vertex_pair_is_adjacent
  validate edge_count_matches_n_choose_2
  validate degree_sequence_equals_[n-1 repeated n times]
  validate adjacency_matrix_symmetric_zero_diagonal_all_other_entries_one
  boundary_tests: K_0, K_1, K_2, K_3, K_4, K_5
  counterexample: path_on_three_vertices is not complete
  counterexample: triangle_with_one_missing_edge is not complete

8. SEMANTIC_EQUIVALENCE
  complete_simple_undirected_graph <=> pairwise_adjacent_distinct_vertices
  complete_simple_undirected_graph_of_order_n <=> K_n up to isomorphism
  complete_simple_undirected_graph_of_order_n <=> simple_graph_with_n(n-1)/2_edges
  complete_simple_undirected_graph_of_order_n <=> simple_graph_with_all_degrees_n-1
  NOTE: These equivalences require the specified simple undirected graph assumptions.

9. CONFIDENCE
  mathematical_definitions: 1.00
  standard_theorems: 1.00
  live_source_alignment: unverified
  interpretation_policy: Distinguish proved graph properties from empirical claims.

10. FLOW_HASH
  algorithm: SHA-256
  scope: canonicalised ontology text excluding mutable hash field
  value: compute_at_ingestion
  note: Do not invent a digest before canonicalisation and hashing.

11. UPDATE_MODEL
  update_trigger: correction_to_definition | theorem | validation | source_page
  version_policy: increment version; preserve provenance and prior revision
  dependency_checks: GraphOntology, VertexOntology, EdgeOntology, CliqueOntology
  revalidate: algebraic laws, boundary cases, semantic equivalences

12. AI_CONSUMPTION
  parse_as: typed graph ontology with declarative predicates and rules
  primary_entity: CompleteGraph
  canonical_notation: K_n
  query_examples:
    - Is K_5 planar? -> false
    - How many edges in K_6? -> 15
    - What is the chromatic number of K_4? -> 4
    - Is K_2 bipartite? -> true
  guardrails:
    - Apply simple-graph assumptions before using complete-graph identities.
    - Do not confuse complete graphs with complete bipartite graphs.
    - Respect empty and singleton graph conventions.
    - Do not claim that a source page was verified without retrieval.

END FLOW ALGEBRA