Results 1 to 2 of 2

Math Help - Graph Theory - Size of a Line Graph

  1. #1
    Newbie
    Joined
    Jul 2010
    Posts
    2

    Graph Theory - Size of a Line Graph

    Hi, all!

    Line graph L(G) of a graph G is a graph in which every edge in E(G) is represented with a vertex. Two vertices in L(G) are adjacent if and only if the corresponding edges in G share a vertex.

    Now suppose graph G has n vertices, labeled v_1, v_2, \dots, v_n and the degree of each vertex is deg(v_i) = r_i.

    Find the size of L(G).

    I have attempted to solve it, but I'm stuck.

    The order of L(G) is \frac{1}{2} \sum^{n}_{i=1} r_i. Let's call it m.
    The vertex v \in V(L(G)) is the edge from some vertex u_i \in V(G) to another u_k \in V(G), it's degree is therefore deg(v_j) = (deg(u_{i_j}) - 1) + (deg(u_{k_j}) - 1) = r_{i_j} + r_{k_j} - 2.
    Size of L(G) = \frac{1}{2} \sum^{m}_{j=1}(r_{i_j}+r_{k_j}) - m

    While I'm quite sure this is correct, it doesn't seem to be very useful - or indeed the expected answer. Any help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jul 2010
    Posts
    2
    Okay, I think I got now. If you write out the terms, you see that each r_i is repeated exactly r_i times. We can then write the size of L(G) as |E(L(G))| = \frac{1}{2}\sum^{n}_{i=1}{r_i^2} - m.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Graph theory, bipartite Graph
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: March 10th 2012, 06:47 AM
  2. Graph Theory / Chromatic Number of a Complete Graph
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 15th 2011, 10:59 AM
  3. (Graph Theory)Prove that graph X is a tree.
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: August 1st 2011, 04:30 AM
  4. Replies: 0
    Last Post: September 25th 2010, 06:59 AM
  5. Turning an exponential graph into a line graph
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: February 9th 2008, 06:16 PM

Search Tags


/mathhelpforum @mathhelpforum