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
AABBCCDD0110101011010010

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

deg(D) =
2.Easy

A simple graph has 7 vertices with degrees 4, 3, 3, 3, 2, 2, 1. How many edges |E| does it have?

deg: 4, 3, 3, 3, 2, 2, 1
|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 19 vertices and 22 edges. How many edges k must be removed to leave a spanning tree?

|V| = 19|E| = 22
k =
5.Hard
310655101211111215ABCDEFG

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
ABCDE

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

M = adjacency matrixL = 2
N =

Answer key

  1. 1. deg(D) = 1
  2. 2. |E| = 9
  3. 3. Euler path, no circuit
  4. 4. k = 4
  5. 5. d(F) = 19
  6. 6. N = 2

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