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 34 vertices and 3 connected components. How many edges |E| does it have?

|V| = 34c = 3
|E| =
2.Easy

A tree has 14 edges. How many vertices |V| does it have?

|E| = 14
|V| =
3.Medium

How many different Hamilton circuits H does the complete graph K7 have? Circuits that differ only in the starting vertex or the direction count as the same.

K7
H =
4.Medium

A tree has 3 vertices of degree 3, 4 vertices of degree 4, 3 vertices of degree 5, 3 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

3 × deg 34 × deg 43 × deg 53 × deg 2
L =
5.Hard
714713115714ABCDEF

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
ABCDEF

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. |E| = 31
  2. 2. |V| = 15
  3. 3. H = 360
  4. 4. L = 22
  5. 5. d(F) = 27
  6. 6. N = 3

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