27 exam-style questions on Graph theory: graphs, vertices, edges, degree, trees — topic 3.14 of Geometry and Trigonometry in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
A tree has 15 vertices. State the number of edges it contains.
Answer: 14
Any tree on n vertices has exactly n − 1 edges, so 15 − 1 = 14.
A simple connected graph has exactly five vertices, with degrees 2, 3, 3, 4 and 4. Calculate the number of edges.
Answer: 8
By the handshaking lemma the sum of degrees (2+3+3+4+4 = 16) equals twice the number of edges, so edges = 16 ÷ 2 = 8.
A graph has degree sequence 4, 3, 3, 2, 1, 1. Determine whether it has an Eulerian trail.
Answer: No, because there are 4 odd-degree vertices
The odd-degree vertices are those with degree 3, 3, 1, 1 — that is 4 of them. An Eulerian trail requires 0 or exactly 2 odd-degree vertices, so none exists.
The complete graph K₇ has every pair of its 7 vertices joined by exactly one edge. Calculate the total number of edges.
Answer: 21
The number of edges of Kₙ is n(n − 1)/2, so 7 × 6 ÷ 2 = 21.
State which of the following statements about a tree is always true.
Answer: A tree is connected and has no cycles
A tree is by definition a connected graph with no cycles; consequently it has n − 1 edges, not n.
A weighted connected graph has 6 vertices. Determine the number of edges in any of its minimum spanning trees.
Answer: 5
A spanning tree of a graph with n vertices always has n − 1 edges, so 6 − 1 = 5, regardless of the weights.
That was 6 of 27 Graph theory: graphs, vertices, edges, degree, trees questions.
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27 Graph theory: graphs, vertices, edges, degree, trees questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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