# Finite Simple Undirected Graph Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261003-advance-20261004-r04-graph-foundations · 2026-10-03 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V=∅ and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {1, 0} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0001/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 2. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-11, -11} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0002/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 3. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-10, -5} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0003/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 4. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-8, -9} an edge in E?

View answer and explanationYes

Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0004/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 5. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-8, -6} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0005/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 6. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-7, -3} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0006/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 7. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -6} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0007/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 8. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-5, 0} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0008/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 9. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-4, -1} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0009/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 10. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -3} an edge in E?

View answer and explanationYes

Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0010/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 11. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, 2} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0011/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 12. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-1, 5} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0012/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 13. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {2, 0} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0013/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 14. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {1, 4} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0014/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 15. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {2, 7} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0015/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 16. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {4, 3} an edge in E?

View answer and explanationYes

Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0016/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 17. Edge Membership in a Simple Undirected Graph

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. Is the candidate endpoint set {4, 6} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0017/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 18. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {5, 9} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0018/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 19. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 6} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0019/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 20. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {7, 12} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0020/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 21. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {8, 11} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0021/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 22. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 9} an edge in E?

View answer and explanationYes

Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0022/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 23. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 14} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0023/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 24. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {11, 17} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0024/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 25. Edge Membership in a Simple Undirected Graph

The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {14, 12} an edge in E?

View answer and explanationNo

Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0025/)

Finite simple undirected graphs · qf-simple-graph-edge-membership · candidate





## 26. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={0} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=0 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(0)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0026/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 27. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-9 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-10, -8}

Find distinct neighbors using the explicit edge set: N(-9)={-10, -8}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0027/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 28. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-7 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-10}

Find distinct neighbors using the explicit edge set: N(-7)={-10}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0028/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 29. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(-5)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0029/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 30. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-6 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(-6)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0030/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 31. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-7, -6, -4}

Find distinct neighbors using the explicit edge set: N(-5)={-7, -6, -4}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0031/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 32. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(-4)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0032/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 33. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-3 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-4, -2}

Find distinct neighbors using the explicit edge set: N(-3)={-4, -2}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0033/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 34. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-3}

Find distinct neighbors using the explicit edge set: N(-4)={-3}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0034/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 35. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-3, -2}

Find distinct neighbors using the explicit edge set: N(-1)={-3, -2}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0035/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 36. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{-2}

Find distinct neighbors using the explicit edge set: N(-1)={-2}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0036/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 37. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{0, 1, 2, 3, 4}

Find distinct neighbors using the explicit edge set: N(-1)={0, 1, 2, 3, 4}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0037/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 38. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=1 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(1)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0038/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 39. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=3 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{2, 4}

Find distinct neighbors using the explicit edge set: N(3)={2, 4}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0039/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 40. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=2 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{3, 4, 5}

Find distinct neighbors using the explicit edge set: N(2)={3, 4, 5}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0040/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 41. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(7)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0041/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 42. Neighbors in a Simple Undirected Graph

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. The vertex v=6 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(6)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0042/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 43. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{5, 6, 8}

Find distinct neighbors using the explicit edge set: N(7)={5, 6, 8}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0043/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 44. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=10 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(10)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0044/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 45. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=9 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{8, 10}

Find distinct neighbors using the explicit edge set: N(9)={8, 10}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0045/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 46. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=8 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{9}

Find distinct neighbors using the explicit edge set: N(8)={9}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0046/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 47. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{9, 10}

Find distinct neighbors using the explicit edge set: N(11)={9, 10}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0047/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 48. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{12}

Find distinct neighbors using the explicit edge set: N(13)={12}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0048/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 49. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation{12, 13, 14, 15, 16}

Find distinct neighbors using the explicit edge set: N(11)={12, 13, 14, 15, 16}. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0049/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 50. Neighbors in a Simple Undirected Graph

The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. List N(v) in increasing order with no duplicates.

View answer and explanation∅

Find distinct neighbors using the explicit edge set: N(13)=∅. Do not infer adjacency from a sketch or drawn-line crossings.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0050/)

Finite simple undirected graphs · qf-simple-graph-neighbors · candidate





## 51. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={0} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=0 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(0)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0051/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 52. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-9 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(-9)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0052/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 53. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-7 belongs to V. Find deg(v).

View answer and explanation1

Count distinct neighbors from the explicit edge set to obtain deg(-7)=1. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0053/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 54. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(-5)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0054/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 55. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-6 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(-6)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0055/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 56. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. Find deg(v).

View answer and explanation3

Count distinct neighbors from the explicit edge set to obtain deg(-5)=3. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0056/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 57. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(-4)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0057/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 58. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-3 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(-3)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0058/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 59. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. Find deg(v).

View answer and explanation1

Count distinct neighbors from the explicit edge set to obtain deg(-4)=1. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0059/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 60. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(-1)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0060/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 61. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).

View answer and explanation1

Count distinct neighbors from the explicit edge set to obtain deg(-1)=1. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0061/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 62. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).

View answer and explanation5

Count distinct neighbors from the explicit edge set to obtain deg(-1)=5. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0062/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 63. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=1 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(1)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0063/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 64. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=3 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(3)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0064/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 65. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=2 belongs to V. Find deg(v).

View answer and explanation3

Count distinct neighbors from the explicit edge set to obtain deg(2)=3. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0065/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 66. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(7)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0066/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 67. Vertex Degree in a Simple Undirected Graph

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. The vertex v=6 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(6)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0067/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 68. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. Find deg(v).

View answer and explanation3

Count distinct neighbors from the explicit edge set to obtain deg(7)=3. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0068/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 69. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=10 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(10)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0069/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 70. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=9 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(9)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0070/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 71. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=8 belongs to V. Find deg(v).

View answer and explanation1

Count distinct neighbors from the explicit edge set to obtain deg(8)=1. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0071/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 72. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. Find deg(v).

View answer and explanation2

Count distinct neighbors from the explicit edge set to obtain deg(11)=2. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0072/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 73. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. Find deg(v).

View answer and explanation1

Count distinct neighbors from the explicit edge set to obtain deg(13)=1. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0073/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 74. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. Find deg(v).

View answer and explanation5

Count distinct neighbors from the explicit edge set to obtain deg(11)=5. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0074/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 75. Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. Find deg(v).

View answer and explanation0

Count distinct neighbors from the explicit edge set to obtain deg(13)=0. An isolated vertex has degree zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0075/)

Finite simple undirected graphs · qf-simple-graph-degree · candidate





## 76. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V=∅ and E=∅. 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.

View answer and explanation0

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0076/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 77. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0077/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 78. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0078/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 79. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0079/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 80. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. 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.

View answer and explanation2

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0080/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 81. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. 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.

View answer and explanation12

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=12. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0081/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 82. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. 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.

View answer and explanation0

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0082/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 83. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. 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.

View answer and explanation10

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=10. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0083/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 84. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. 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.

View answer and explanation2

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0084/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 85. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0085/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 86. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. 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.

View answer and explanation4

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=4. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0086/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 87. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. 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.

View answer and explanation30

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=30. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0087/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 88. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={0, 1, 2} and E=∅. 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.

View answer and explanation0

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0088/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 89. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0089/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 90. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0090/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 91. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0091/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 92. Total Vertex Degree in a Simple Undirected Graph

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.

View answer and explanation2

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0092/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 93. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. 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.

View answer and explanation12

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=12. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0093/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 94. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. 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.

View answer and explanation0

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0094/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 95. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. 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.

View answer and explanation10

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=10. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0095/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 96. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. 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.

View answer and explanation2

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0096/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 97. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. 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.

View answer and explanation6

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0097/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 98. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. 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.

View answer and explanation4

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=4. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0098/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 99. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. 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.

View answer and explanation30

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=30. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0099/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate





## 100. Total Vertex Degree in a Simple Undirected Graph

The finite simple undirected graph has V={12, 13, 14} and E=∅. 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.

View answer and explanation0

Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r04-graph-0100/)

Finite simple undirected graphs · qf-simple-graph-handshaking · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r04-graph-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r04-graph-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r04-graph-foundations/validation-report.json)