# Question 8002: Complete Adjacency Matrix, Degrees, and Handshake Counts

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0015-08002 · q-20261012-advance-20261017-r05-graph-0002



## Original question and givens

Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).


- G001 — Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-graph](https://ocme.evemisslab.com/en/mko/mko-graph/)





## Step-by-step derivation

S001 · impliesBefore

n=1;raw edges=[]

After

canonical edges=[]

Conditions: none

Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing.

Step identifiers and dependenciesRule: simple_graph_edge_normalization · Uses: G001





S002 · impliesBefore

phase=original;vertex=0;all edges=[]

After

neighbors=[];degree=0

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S001





S003 · impliesBefore

vertex=0;ordered targets=[0];neighbors=[]

After

adjacency row=[No]

Conditions: none

Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.

Step identifiers and dependenciesRule: complete_boolean_adjacency_row · Uses: G001, S002





S004 · calculationBefore

degree sum := sum(0)

After

degree sum = 0

Conditions: none

Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

twice distinct edge count := mul(2, 0)

After

twice distinct edge count = 0

Conditions: none

Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

degrees=[0];edge count=0

After

sum=0;2|E|=0

Conditions: none

Each undirected nonloop edge contributes1 at each endpoint,so the degree sum is twice the canonical edge count.This does not test connectedness or treehood.

Step identifiers and dependenciesRule: handshake_each_edge_two_ends · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute twice distinct edge count with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute degree sum with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(1, [], [[No]], [0], 0, 0)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: ccec157f81ece99465f7b78bf5f1c1a76eb2bad9fed525ead3aa552a959914f7
- candidate record SHA-256: 682faba1698f84d17998651174cc9b5ad070ec420f807113fe1ce05a88282939
- semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
- question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r05-graph-0002.json)