Geodesic
Wiener index of generalized stars and their quadratic line graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 161-175

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The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property are constructed.
Keywords: distance in a graph, Wiener index, star, iterated line graph 
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Dobrynin, Andrey; Mel'nikov, Leonid. Wiener index of generalized stars and their quadratic line graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 161-175. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a14/