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

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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.
Classification : 05C05, 05C12
Mots-clés : distance in a graph, Wiener index, star, iterated line graph, line graph 
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Andrey A. Dobrynin; Leonid S. Mel'nikov. Wiener index of generalized stars and their quadratic line graphs. Discussiones Mathematicae Graph Theory, Tome 26 (2006) no. 1, p. 161. https://geodesic-test.mathdoc.fr/item/DMGT_2006__26_1_270582/