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 9 vertices and 5 connected components. How many edges |E| does it have?

|V| = 9c = 5
|E| =
2.Easy
AABBCCDD0111101011011010

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

deg(A) =
3.Medium

A tree has 4 vertices of degree 3, 3 vertices of degree 4, 4 vertices of degree 5, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 33 × deg 44 × deg 5
L =
4.Medium
ABCDEFGEuler 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| = 7|E| = 10
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
14811189214ABCDEF

Use Dijkstra’s algorithm to find the length d(E) of the shortest path from A to E in the weighted graph.

start: Aend: E
d(E) =
6.Hard
81016101018ABCDEF

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) =

Answer key

  1. 1. |E| = 4
  2. 2. deg(A) = 3
  3. 3. L = 24
  4. 4. neither
  5. 5. d(E) = 25
  6. 6. d(F) = 17

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