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 simple graph has 5 vertices with degrees 3, 3, 3, 2, 1. How many edges |E| does it have?

deg: 3, 3, 3, 2, 1
|E| =
2.Easy

A 3-regular graph has 6 vertices (every vertex has degree 3). How many edges |E| does it have?

n = 6k = 3
|E| =
3.Medium

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

3 × deg 42 × deg 2
L =
4.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| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
ABCDE

How many walks of length 2 go from B to C? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =
6.Hard
101213123101810ABCDEFG

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 4; A itself is round 1.

start: Around 4
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
  7. 7G

Answer key

  1. 1. |E| = 6
  2. 2. |E| = 9
  3. 3. L = 8
  4. 4. Euler circuit
  5. 5. N = 1
  6. 6. D

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