Wiener index of generalized stars and their quadratic line graphs

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 ∆ ≥ 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 ≤ ∆ ≤ 6. Infinite families of generalized stars with this property are constructed.