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 18 vertices and 3 connected components. How many edges |E| does it have?

|V| = 18c = 3
|E| =
2.Easy

A simple graph has 5 vertices with degrees 4, 3, 3, 2, 2. How many edges |E| does it have?

deg: 4, 3, 3, 2, 2
|E| =
3.Medium
ABCDEEuler 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| = 5|E| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

A connected graph has 18 vertices and 29 edges. How many edges k must be removed to leave a spanning tree?

|V| = 18|E| = 29
k =
5.Hard
ABCDEF

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

M = adjacency matrixL = 3
N =
6.Hard
15196131314ABCDE

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

|V| = 5|E| = 7
W =

Answer key

  1. 1. |E| = 15
  2. 2. |E| = 7
  3. 3. Euler path, no circuit
  4. 4. k = 12
  5. 5. N = 6
  6. 6. W = 25

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