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Wiener index of line graphs
Knor, Martin ; Škrekovski, Riste
We consider the molecular descriptor Wiener index, ▫$W$▫, of graphs and their line graphs. This index plays a crucial role in organic chemistry. It was studied by chemists decades before it attracted ...attention of mathematicians. In fact, it was studied long time before the branch of discrete mathematics, which is now known as graph theory, was born. Nowadays. there are many indices known used to describe the molecules. In this chapter, we first introduce the concept of topological indices and list some of them. Next, in the third section, we focus on the Wiener index and expose some of its properties. Furthermore, we compare the values ▫$W(G)$▫ and ▫$W(L(G))$▫, in particular when they are equal for ▫$G$▫ being in various classes of graphs. In addition, we expose some bounds of the Wiener index of the line graph in terms of the Gutman index of the original graph. In the next section, we consider the equality ▫$W(G) = W(L(G))$▫ for graphs with large ginh. Finally, we consider the same equality for trees but for higher iterations of line graphs, ▫$W(T) = W^i(L(T))$▫. The sixth section is dedicated to the case ▫$i = 2$▫. In the seventh section, we show that a solution of ▫$$W^i(L(T)) = W(T) \quad (i \ge 3)$$▫ exists only for▫ $i = 3$▫, and it is one particular class of trees, all homeomorphic to the letter ▫$H$▫. The smallest such tree has 388 vertices.
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