# GATE | GATE CS Mock 2018 | Question 33

The line graph L(G) of a simple graph G is defined as follows:

• There is exactly one vertex v(e) in L(G) for each edge e in G.
• For any two edges e and e’ in G, L(G) has an edge between v(e) and v(e’), if and only if e and e’are incident with the same vertex in G.

Which of following option is not correct about “Line Graph”?
(A) A line graph has an articulation point if and only if the underlying graph has a bridge for which neither endpoint has degree one
(B) For a graph G with n vertices and m edges, the number of vertices of the line graph L(G) is m, and the number of edges of L(G) is half the sum of the squares of the degrees of the vertices in G, m.
(C) If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the line graph of G is Hamiltonian.
(D) None of these

Explanation: For a graph G with n vertices and m edges, the number of vertices of the line graph L(G) is m, and the number of edges of L(G) is half the sum of the squares of the degrees of the vertices in G, minus m.

Option (B) is false.

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