Question 8005: Complete Adjacency Matrix, Degrees, and Handshake Counts

Original question and givens

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. 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(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. 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

Step-by-step derivation

n=4;raw edges=[[0, 1], [1, 2], [2, 3]]

canonical edges=[[0, 1], [1, 2], [2, 3]]

phase=original;vertex=0;all edges=[[0, 1], [1, 2], [2, 3]]

neighbors=[1];degree=1

phase=original;vertex=1;all edges=[[0, 1], [1, 2], [2, 3]]

neighbors=[0, 2];degree=2

phase=original;vertex=2;all edges=[[0, 1], [1, 2], [2, 3]]

neighbors=[1, 3];degree=2

phase=original;vertex=3;all edges=[[0, 1], [1, 2], [2, 3]]

neighbors=[2];degree=1

vertex=0;ordered targets=[0, 1, 2, 3];neighbors=[1]

adjacency row=[No, Yes, No, No]

vertex=1;ordered targets=[0, 1, 2, 3];neighbors=[0, 2]

adjacency row=[Yes, No, Yes, No]

vertex=2;ordered targets=[0, 1, 2, 3];neighbors=[1, 3]

adjacency row=[No, Yes, No, Yes]

vertex=3;ordered targets=[0, 1, 2, 3];neighbors=[2]

adjacency row=[No, No, Yes, No]

degree sum := sum(1, 2, 2, 1)

degree sum = 6

twice distinct edge count := mul(2, 3)

twice distinct edge count = 6

degrees=[1, 2, 2, 1];edge count=3

sum=6;2|E|=6

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute twice distinct edge count with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute degree sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(4, [[0, 1], [1, 2], [2, 3]], [[No, Yes, No, No], [Yes, No, Yes, No], [No, Yes, No, Yes], [No, No, Yes, No]], [1, 2, 2, 1], 6, 6)

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.