Wiener Index -- from Wolfram MathWorld

The Wiener index , denoted (Wiener 1947) and also known as the "path number" or Wiener number (Plavšić et al. 1993), is a graph index defined for a graph on nodes by

where is the graph distance matrix.

Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon (Devillers and Balaban 1999, p. 25).

The Wiener index of a graph with vertex count is related to the average disorder number of the graph by

(Fried 2022).

The Wiener index of a graph Cartesian product of graphs and is given by

(Yeh and Gutman 1994, Fried 2022).

The Wiener index is not very discriminant. In fact, the paw graph and square graph on four nodes are already indistinguishable using the Wiener index (both have value 8). The numbers of non-Wiener-unique connected graphs on , 2, ... nodes given by 0, 0, 0, 2, 16, 108, 847, 11110, 261072, ... (OEIS A193217).

Precomputed values for many graphs are implemented in the Wolfram Language as GraphData[g, "WienerIndex"].

The following table summarizes values of the Wiener index for various special classes of graphs.

Closed forms are summarized in the following table. The cycle graph was considered by Plavšić et al. (1993) and Babić et al. (2002) and the path graph by Plavšić et al. (1993).

See also

Average Disorder Number, Balaban Index, Graph Distance Matrix, Kirchhoff Index, Resistance Distance, Topological Index, Wiener Sum Index

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References

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90, 166-176, 2002.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 26 and 108-109, 1999.Entringer, R. C.; Jackson, D. E.; and Snyder, D. "Distance in Graphs." Czech. Math. J. 26, 283-296, 1976.Fried, S. "The Disorder Number of a Graph." 7 Aug 2022. https://arxiv.org/abs/2208.03788/.Hosoya, H. "Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons." Bull. Chem. Soc. Japan 44, 2322-2239, 1971.Plavšić, D.; Nikolić, S.; Trinajstić, N.; and Mihalić, Z. "On the Harary Index for the Characterization of Chemical Graphs." J. Math. Chem. 12, 235-250, 1993.Sloane, N. J. A. Sequence OEIS A193217 in "The On-Line Encyclopedia of Integer Sequences."Wiener, H. J. "Structural Determination of Paraffin Boiling Points." J. Amer. Chem. Soc. 69, 17-20, 1947.Wiener, H. "Influence of Interatomic Forces on Paraffin Properties." J. Chem. Phys. 15, 766, 1947.Wiener, H. "Vapor Pressure-Temperature Relationships Among the Branched Paraffin Hydrocarbons." J. Phys. Chem. 52, 425-430, 1948.Wiener, H. "Relation of the Physical Properties of the Isomeric Alkanes to Molecular Structure. Surface Tension, Specific Dispersion, and Critical Solution Temperature in Aniline." J. Phys. Chem. 52, 1082-1089, 1948.Yeh, Y.-N. and Gutman, I. "On the Sum of All Distances in Composite Graphs." Disc. Math. 135, 359-365, 1994.Zerovnik, J. "Szeged Index of Symmetric Graphs." J. Chem. Inf. Comput. Sci. 39, 77-80, 1999.

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Wiener Index

Cite this as:

Weisstein, Eric W. "Wiener Index." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WienerIndex.html

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