# Question 35267: Total Vertex Degree in a Simple Undirected Graph

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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.

R0069-35267 · q-20261003-advance-20261004-r04-graph-0092



## Original question and givens

The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.


- G001 — The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.





## 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

source roles = [qf-simple-graph-handshaking, [E, V, operation]]

After

derived = [handshaking, no-invented-domain]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependenciesRule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001





S002 · impliesBefore

source roles = [LDL graph vertices ORIGINAL, 100, [4, 5, 6]]

After

derived = [[4, 5, 6]]

Conditions: none

Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.

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





S003 · impliesBefore

source roles = [LDL raw edges ORIGINAL, 0, [4, 5]]

After

derived = [[4, 5]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

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





S004 · impliesBefore

source roles = [[4, 5, 6], [[4, 5]], No]

After

derived = [[[4, 5]], [[4, [5]], [5, [4]], [6, []]]]

Conditions: none

Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.

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





S005 · impliesBefore

source roles = [4, [5]]

After

derived = [1]

Conditions: none

For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.

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





S006 · impliesBefore

source roles = [5, [4]]

After

derived = [1]

Conditions: none

For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.

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





S007 · impliesBefore

source roles = [6, []]

After

derived = [0]

Conditions: none

For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.

Step identifiers and dependenciesRule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S006





S008 · calculationBefore

Graph ORIGINAL all vertex degrees sum := sum(1, 1, 0)

After

Graph ORIGINAL all vertex degrees sum = 2

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

Graph ORIGINAL two endpoints per normalized edge := mul(2, 1)

After

Graph ORIGINAL two endpoints per normalized edge = 2

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [[4, 5, 6], [[4, 5]], [1, 1, 0]]

After

derived = [2, 2]

Conditions: none

Every undirected edge contributes TWO endpoint incidences,so the COMPLETE degree sum is2|E|.For the empty graph both sides are0.This is finite full recomputation/double counting,not a sketch inference.

Step identifiers and dependenciesRule: Graph_HANDSHAKE_complete_incidence_double_count · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S009 · premise_and_result_audit — Recompute Graph ORIGINAL two endpoints per normalized edge with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute Graph ORIGINAL all vertex degrees sum with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S006 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S005 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S003 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.





## Result

2

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: eeaffd15c434547bad00330b3ca035137f372e6bb2653620ebcfa9079b44701d
- candidate record SHA-256: d14391668d4443160d8e4fb6d9ffec3202a7eb712881c7f456d01cc065426eea
- semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
- question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r04-graph-0092.json)