Graph theory worksheet with answers

Free printable Graph theory worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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Name: ____________________   Date: ____________

1.Easy

A 5-regular graph has 8 vertices (every vertex has degree 5). How many edges |E| does it have?

n = 8k = 5
|E| =
2.Easy

A tree has 22 edges. How many vertices |V| does it have?

|E| = 22
|V| =
3.Medium

How many different Hamilton circuits H does the complete graph K4 have? Circuits that differ only in the starting vertex or the direction count as the same.

K4
H =
4.Medium

A tree has 3 vertices of degree 3, 2 vertices of degree 4, 2 vertices of degree 5, 4 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

3 × deg 32 × deg 42 × deg 54 × deg 2
L =
5.Hard
6912104101081ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 5; A itself is round 1.

start: Around 5
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
6.Hard
ABCDE

How many closed walks of length 3 start and end at E? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 3
N =

Answer key

  1. 1. |E| = 20
  2. 2. |V| = 23
  3. 3. H = 3
  4. 4. L = 15
  5. 5. F
  6. 6. N = 4

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