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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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 simple graph has 7 vertices with degrees 4, 4, 4, 3, 2, 2, 1. How many edges |E| does it have?

deg: 4, 4, 4, 3, 2, 2, 1
|E| =
3.Medium

How many different Hamilton circuits H does the complete graph K6 have? Circuits that differ only in the starting vertex or the direction count as the same.

K6
H =
4.Medium

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

4 × deg 32 × deg 5
L =
5.Hard
ABCDEF

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

M = adjacency matrixL = 3
N =
6.Hard
15110416597ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

Answer key

  1. 1. |E| = 20
  2. 2. |E| = 10
  3. 3. H = 60
  4. 4. L = 12
  5. 5. N = 4
  6. 6. d(F) = 13

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