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 tree has 26 vertices. How many edges |E| does it have?

|V| = 26
|E| =
2.Easy

How many edges |E| does the complete bipartite graph K2,12 have?

K2,12
|E| =
3.Medium

A tree has 3 vertices of degree 3, 2 vertices of degree 4, 3 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 43 × deg 54 × 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
ABCDEF

How many closed walks of length 2 start and end at E? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =
6.Hard
8181416913ABCDE

Prim’s algorithm builds a minimum spanning tree starting from vertex D. Click the edge it adds as edge number 4.

start: Dk = 4
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge BE

Answer key

  1. 1. |E| = 25
  2. 2. |E| = 24
  3. 3. L = 18
  4. 4. neither
  5. 5. N = 4
  6. 6. edge CE

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