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

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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/