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
ABCDEF

Exactly one vertex of the graph has degree 4. Click it.

deg = 4
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
2.Easy
AABBCCDD0111101011001000

The adjacency matrix of a simple undirected graph is shown. Find the degree deg(B) of vertex B.

deg(B) =
3.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| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

A connected graph has 16 vertices and 22 edges. How many edges k must be removed to leave a spanning tree?

|V| = 16|E| = 22
k =
5.Hard
ABCDEF

How many walks of length 3 go from F to E? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 3
N =
6.Hard
153761151512ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 3; A itself is round 1.

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

Answer key

  1. 1. E
  2. 2. deg(B) = 2
  3. 3. Euler path, no circuit
  4. 4. k = 7
  5. 5. N = 1
  6. 6. B

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