Wiener Sum Index -- from Wolfram MathWorld

The Wiener sum index is a graph index defined for a graph on nodes by

where is the graph distance matrix and is the resistance 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 following table summarizes values of the Kirchhoff sum index for various special classes of graphs.

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

Closed forms for some special classes of graphs are summarized in the table below, where denotes a harmonic number.

See also

Balaban Index, Graph Distance Matrix, Kirchhoff Sum Index, Resistance Distance, Wiener Index

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References

Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, 1999.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.Randić, M. "On Characterization of Cyclic Structures." J. Chem. Inf. Comput. Sci. 37, 1063-1071, 1997.

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Wiener Sum Index

Cite this as:

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

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