Lesson DA6-DA7

DA6-DA7 Trees and isomorphism of graphs Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DA6-DA7, Trees and isomorphism of graphs: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. What is a tree in graph theory?

    • A connected graph with no cycles
    • A graph in which every vertex has degree 2
    • A graph with no vertices of odd degree
    • A graph in which every pair of vertices is joined by an edge

  2. What is a simple connected graph?

    • A connected graph with exactly one cycle
    • A connected graph in which every vertex has degree 1
    • A connected graph in which every edge has a weight
    • A connected graph with no loops and no multiple edges

  3. How many edges does a tree with 9 vertices have?

    • 9
    • 18
    • 8
    • 7

  4. A tree has 5 vertices. One vertex has degree 3, another has degree 2, and the remaining vertices are leaves of degree 1. How many leaves are there?

    • 3
    • 2
    • 1
    • 4

  5. Two simple graphs each have 4 vertices and 3 edges. Which statement is correct?

    • They need not be isomorphic, since a path on 4 vertices and a triangle with an isolated vertex both have 4 vertices and 3 edges
    • They are necessarily isomorphic, since any two graphs with the same number of vertices and edges are isomorphic
    • They cannot both be simple, since one of them must contain a loop
    • They are necessarily isomorphic, since both have the same number of cycles

  6. Which property is preserved by every isomorphism between two graphs?

    • The labels given to the vertices
    • The degree sequence
    • The positions of the vertices in a drawing
    • The names given to the edges

  7. Graph A has 5 vertices and 6 edges. Graph B has 5 vertices and 7 edges. Which statement is correct?

    • They are isomorphic, since both can be drawn without crossings
    • They are isomorphic, since they have the same number of vertices
    • They cannot be isomorphic, since isomorphic graphs have the same number of edges
    • They can be isomorphic if the extra edge in Graph B is treated as a loop

  8. A tree has 7 vertices. If one leaf and its edge are removed, how many edges are left?

    • 6
    • 5
    • 4
    • 7

  9. A connected graph has 6 vertices and 9 edges. How many edges must be removed to leave a spanning tree?

    • 9
    • 4
    • 3
    • 5

  10. A connected graph has v vertices and v - 1 edges. Which kind of graph must it be?

    • A tree
    • A complete graph
    • A cycle
    • A graph containing a loop

  11. Which simple connected graph has four vertices, each of degree 2?

    • The cycle on four vertices
    • The path on four vertices
    • The star with four leaves
    • The complete graph K4

  12. Which tree on five vertices has a single vertex of degree 4?

    • The complete bipartite graph K2,3
    • The cycle on five vertices
    • The star K1,4
    • The path on five vertices

  13. A tree has 8 vertices. Two vertices have degree 3, two vertices have degree 2, and all the other vertices are leaves. How many leaves does the tree have?

    • 5
    • 4
    • 3
    • 2

  14. Which degree sequence could belong to a tree on six vertices?

    • 2, 2, 2, 2, 2, 2
    • 4, 4, 1, 1, 1, 1
    • 3, 3, 1, 1, 1, 1
    • 2, 2, 2, 2, 2, 1

  15. Which statement about isomorphism of graphs is correct?

    • Isomorphic graphs share vertex count, edge count and degree sequence, but matching these alone does not prove isomorphism
    • Two graphs with the same degree sequence are always isomorphic
    • Isomorphic graphs must be drawn with the same number of edge crossings
    • Two graphs are isomorphic if one contains a cycle and the other does not

  16. How many non-isomorphic trees are there on four vertices?

    • 2
    • 1
    • 4
    • 3

  17. How many non-isomorphic trees are there on five vertices?

    • 4
    • 3
    • 2
    • 5

  18. Which pair of graphs is definitely not isomorphic, and why?

    • A path on four vertices and a cycle on four vertices, since they have different numbers of edges
    • Two graphs with the same degree sequence, since equal degree sequences always mean the graphs are isomorphic
    • A cycle on four vertices and the complete graph K4, since both are simple and connected
    • Two trees on four vertices, since all trees with the same number of vertices are isomorphic

  19. A tree has 10 vertices. A second connected graph also has 10 vertices and 10 edges. Which statement is correct?

    • The second graph must be a tree, since both graphs are connected
    • The second graph must contain at least one cycle, since a connected graph with 10 vertices and 10 edges is not a tree
    • The second graph cannot contain a cycle, since it is connected
    • The second graph has exactly two spanning trees, regardless of its structure

  20. How many spanning trees does the cycle graph C4 have?

    • 2
    • 3
    • 4
    • 1

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