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For an undirected graph $G=(V, E)$, the line graph $G'=(V', E')$ is obtained by replacing each edge in $E$ by a vertex, and adding an edge between two vertices in $V'$ if the corresponding edges in $G$ are incident on the same vertex. Which of the following is TRUE of line graphs?

  1. the line graph for a complete graph is complete
  2. the line graph for a connected graph is connected
  3. the line graph for a bipartite graph is bipartite 
  4. the maximum degree of any vertex in the line graph is at most the maximum degree in the original graph
  5. each vertex in the line graph has degree one or two
asked in Graph Theory by Veteran (75.6k points)   | 118 views
Option B). is true.
yes, Agreed. (y)
why not option C?
you can check with K(2,2). The line graph obtained is not biparitite.

1 Answer

+2 votes

The line graph of a connected graph is connected. If G is connected, it contains a path connecting any two of its edges, which translates into a path in L(G) containing any two of the vertices of L(G). Therefore, option B is correct.

We can also do this question using elimination of options. 

You can view the following google drive link for the example - https://drive.google.com/open?id=0B1OKeqz0MEWwNTh4czgzalVDNGM

answered by Active (1k points)  
edited by
but your L(G2): e1---e2 it is bipertite right?e1 may be in one partition and e2 on another...it is bipertite i guess.
Hi,

I have corrected the example, and updated the file.
great example you have updated...proving line of tree is not a tree too :p
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