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 forest (a graph without cycles) has 22 vertices and 4 connected components. How many edges |E| does it have?

|V| = 22c = 4
|E| =
2.Easy
AABBCCDD0111101011011010

The adjacency matrix of a simple undirected graph is shown. How many edges |E| does the graph have?

|E| =
3.Medium

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

2 × deg 52 × deg 2
L =
4.Medium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
21323541388ABCDEFG

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) =
6.Hard
161193719125142ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 10
W =

Answer key

  1. 1. |E| = 18
  2. 2. |E| = 5
  3. 3. L = 8
  4. 4. Euler circuit
  5. 5. d(F) = 9
  6. 6. W = 51

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