asst5 - MATH 239 Fall 2011 Assignment 5 DUE NOON Friday 28...

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MATH 239 — Fall 2011 Assignment 5 DUE: NOON Friday 28 October 2011 in the drop boxes opposite the Math Tutorial Centre MC 4067 or next to the St. Jerome’s library for the St. Jerome’s section. 1. Given a graph G , the line graph L ( G ) is defined in the following way: V ( L ( G )) = E ( G ) , E ( L ( G )) = {{ e 1 , e 2 } | | e 1 e 2 | = 1 } . Prove that the line graph L ( K m,n ) to the complete bipartite graph K m,n is regular and find the common degree to all its vertices. SOLUTION. Let ( A, B ) be a bipartition of V ( K m,n ), with | A | = m and | B | = n . Let e V ( L ( K m,n )) = E ( K m,n ). We have e = { a, b } for some a A and some b B . Since { a, b } E ( K m,n ) and { a , b } E ( K m,n ) for all b B and a A , and since those are all the edges containing a and b , we have { e E ( K m,n ) | | e e | = 1 } = { a, b } | b B \ { b } { a , b } | a A \ { a } . Hence deg( e ) = | B | 1 + | A | 1 = m + n 2. Since that result is independent of e , L ( K m,n ) is ( m + n 2)-regular. 2. A sequence of decreasing integers is called graphic if it corresponds to the degrees of the vertices a graph. Which of these sequences are graphic? Justify your answer.
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