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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 ,thel inegraph L ( G )isde±nedinthefo l low ingway : 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 ±nd the common degree to all its vertices. SOLUTION. Let ( A,B )beab ipa r t i t iono f 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 { ° }∈ E ( K m,n )and { a ° ,b E ( K m,n )fo ra l l b ° B and a ° A , and since those are all the edges containing a and b ,wehave { e ° E ( K m,n ) || e ° e | } = ° { ° }| b ° B \{ b } ± ° { a ° 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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asst5 - MATH 239 Fall 2011 Assignment 5 DUE NOON Friday 28...

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