Wiener Index - Example

Example

Butane (C₄H₁₀) has two different structural isomers: n-butane, with a linear structure of four carbon atoms, and isobutane, with a branched structure. The chemical graph for n-butane is a four-vertex path graph, and the chemical graph for isobutane is a tree with one central vertex connected to three leaves.

• The two isomers of butane

• n-Butane

• Isobutane

The n-butane molecule has three pairs of vertices at distance one from each other, two pairs at distance two, and one pair at distance three, so its Wiener index is

The isobutane molecule has three pairs of vertices at distances one from each other (the three leaf-center pairs), and three pairs at distance two (the leaf-leaf pairs). Therefore, its WIener index is

These numbers are instances of formulas for special cases of the Wiener index: it is for any -vertex path graph such as the graph of n-butane, and for any -vertex star such as the graph of isobutane.

Thus, even though these two molecules have the same chemical formula, and the same numbers of carbon-carbon and carbon-hydrogen bonds, their different structures give rise to different Wiener indices.