Geodesic
On a conjecture on the diameter of line graphs of graphs of diameter two
Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 59 .

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Let $F_1$ be the $5$-vertex path, $F_2$ the graph obtained by identifying a vertex of a triangle with one end vertex of the $3$-vertex path and $F_3$ the graph obtained by identifying a vertex of a triangle with a vertex of another triangle. Let $diam(G)$ be the diameter of the graph $G$. In the paper [H. S. Ramane and I. Gutman, {ı Counterexamples for properties of line graphs of graphs of diameter two\/}, Kragujevac J. Math. {\bf 34} (2010), 147-150] it is conjectured that if $diam(G) \leq 2$ and if none of the $F_i$, $i = 1, 2, 3$, is an induced subgraph of $G$, then $diam(L^k(G)) > 2$ for some $k \geq 2$. In this paper we prove this conjecture.
Classification : 05C12 05C75
Keywords: Line graph, diameter (of graph), graph of diameter 2. 
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Harishchandra S. Ramane; Asha B. Ganagi; Ivan Gutman. On a conjecture on the diameter of line graphs of graphs of diameter two. Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 59 . http://geodesic.mathdoc.fr/item/KJM_2012_36_1_a5/