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 2-regular graph has 11 vertices (every vertex has degree 2). How many edges |E| does it have?

n = 11k = 2
|E| =
2.Easy
AABBCCDDEE0110010011100010100101110

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

deg(C) =
3.Medium

A connected graph has 17 vertices and 23 edges. How many edges k must be removed to leave a spanning tree?

|V| = 17|E| = 23
k =
4.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| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
ABCDEF

How many closed walks of length 3 start and end at E? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 3
N =
6.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 =

Answer key

  1. 1. |E| = 11
  2. 2. deg(C) = 2
  3. 3. k = 7
  4. 4. neither
  5. 5. N = 4
  6. 6. N = 5

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