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Math Help - Graph containing n nodes where n is an integer greater than 1.

  1. #1
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    Graph containing n nodes where n is an integer greater than 1.

    Hi

    The following questions i need help on:
    1) Consider a graph containing n cities (nodes), where n is an integer greater than 1.
    a) Give a formula for the number of links (lines in the complete graph.
    b) Give a formula for the number of links (lines) in the minimal spanning tree.

    P.S
    Last edited by mr fantastic; October 23rd 2010 at 10:34 PM. Reason: Re-titled.
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  2. #2
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    (a) Hint: If you draw an edge (link) from each node (n choices) to every other node (n - 1 choices), then you will draw every edge in the complete graph twice.

    (b) Any tree with n nodes has n - 1 edges. By the way, it does not make sense to talk about a "minimal spanning tree" unless the graph has weights (or distances) assigned to edges. This is because any tree is minimal by definition: it ceases to be connected when any edge is removed. A spanning tree can be defined as a minimal set of edges that connect all vertices in a graph.
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  3. #3
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    The number of links (edges)

    Example 10 nodes -

    Node Links
    1 9 Node 1 links to 9 nodes (and not to its self)
    2 8 Node 2 is already linked to bode 1 - so it links to 8 more.
    3 7 Node 3 is already linked to 1 and 2 - so it links to 7 more

    This continues
    4 6
    5 5
    6 4
    7 3
    8 2
    9 1
    10 is already linked to all the others

    So with n nodes - a fully connected graph will have (n^2 - n) / 2 edges.
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  4. #4
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    thanks for the help
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