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Math Help - Trees and their complements, proof by induction

  1. #1
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    Trees and their complements, proof by induction

    Prove using induction that the complement T` of a tree T with n vertices has ((n-1)(n-2))/2 edges, for all integers n>=2

    A complement graph (T`) of T is a graph with the same vertices of T but with connected edges only where the original graph(T) doesn't have edges.

    HINT:Any tree that has more than one vertex has at least one vertex of degree 1.
    I know how induction works but I'm confused as to how to apply it to graphs like trees
    Last edited by jsdieorksw; May 12th 2010 at 07:58 AM. Reason: clarification
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  2. #2
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    Quote Originally Posted by jsdieorksw View Post
    Prove using induction that the complement T` of a tree T with n vertices has ((n-1)(n-2))/2 edges, for all integers n>=2

    A complement graph (T`) of T is a graph with the same vertices of T but with connected edges only where the original graph(T) doesn't have edges.

    HINT:Any tree that has more than one vertex has at least one vertex of degree 1.
    I know how induction works but I'm confused as to how to apply it to graphs like trees

    You have to tell us what definition of a tree you are using. This is important because there are several equivalent definitions.

    Consider the following definition of a tree:

    A graph T with n vertices is a tree if and only if T is connected and T has n-1 edges.

    From the definition I gave above your result follows trivially:

    Then T' must have

    \binom{n}{2} - (n-1) = (n-2)(n-1)/2 edges.
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