Discussiones Mathematicae Graph Theory 26(1) (2006) 161-175
doi: 10.7151/dmgt.1310

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WIENER INDEX OF GENERALIZED STARS AND THEIR QUADRATIC LINE GRAPHS

Andrey A. Dobrynin and Leonid S. Mel'nikov

Sobolev Institute of Mathematics
Russian Academy of Sciences
Siberian Branch, Novosibirsk 630090, Russia
e-mail: dobr@math.nsc.ru, omeln@math.nsc.ru

Abstract

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 and phrases: distance in a graph, Wiener index, star, iterated line graph.

2000 Mathematics Subject Classification: 05C12, 05C05.

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Received 24 June 2005
Revised 22 July 2005