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# Wiener Sum Index

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.

 graph class OEIS , , ... Andrásfai graph A000000/A000000 1, 175/12, 84959/1353, 4384933627811/26124236280, ... antiprism graph A000000/A000000 X, X, 204/5, 532238/6745, 47877676/368433, 906597918/4666585, ... Apollonian network A000000/A000000 12, 2904467/50616, ... black bishop graph A000000/A000000 0, 1, 15, 242259454379/3706922420, ... cocktail party graph A000000/A000000 X, 28/3, 204/5, 744/7, 1960/9, 4260/11, 8148/13, 14224/15, ... complete bipartite graph A000000/A000000 1, 28/3, 171/5, 592/7, 1525/9, 3276/11, 6223/13, 10816/15, ... complete tripartite graph A000000/A000000 9/2, 204/5, 1161/8, 3888/11, 9825/14, 20844/17, 39249/20, ... complete graph A000000/A000000 0, 1, 9/2, 12, 25, 45, 147/2, 112, 162, 225, ... -crossed prism graph A000000/A000000 X, 2344/35, 295518737/1855364, 1143904996/3804075, ... crown graph A000000/A000000 X, X, 111/5, 2344/35, 9145/63, 2924/11, 62629/143, 130768/195, ... cube-connected cycle graph A000000/A000000 X, X, 2373085829482/3510505999, ... cycle graph A000000/A000000 X, X, 9/2, 28/3, 175/12, 111/5, 1813/60, 4264/105, 14391/280, ... Fibonacci cube graph A000000/A000000 1, 3, 593/42, 65264215739843/1291887161565, ... folded cube graph A000000/A000000 1, 12, 592/7, 9920/21, 3237632/1147, 4195744/279, ... gear graph A000000/A000000 X, X, 18905/462, 107212/1479, 211893947/1898424, ... grid graph A000000/A000000 0, 28/3, 107212/1479, ... grid graph A000000/A000000 0, 2344/35, ... halved cube graph A000000/A000000 0, 1, 12, 744/7, 2432/3, 228431520/40703, 253056/7, ... Hanoi graph A000000/A000000 9/2, 17830832037/286489840, ... hypercube graph A000000/A000000 1, 28/3, 2344/35, 2766448/6405, 292437526/112065, ... Keller graph A000000/A000000 X, 9920/21, ... king graph A000000/A000000 0, 12, 65906799847382/641818867935, ... knight graph A000000/A000000 0, X, X, 3736238283778148/12685333796135, ... Menger sponge graph A000000/A000000 647093662/1662595, ... Möbius ladder A000000/A000000 X, X, 171/5, 84959/1353, 859447105/8006068, ... Mycielski graph A000000/A000000 0, 1, 175/12, 18680381340574/124654958285, ... odd graph A000000/A000000 0, 9/2, 100, 9678725/4301, 43454290887/873115, ... pan graph A000000/A000000 X, X, 79/10, 593/42, 8131/396, 11378/385, 9772367/251940, ... path graph A000217 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... permutation star graph A000000/A000000 0, 1, 111/5, 2030300070/2679523, ... prism graph A000000/A000000 X, X, 1435/44, 2344/35, 102133740/954767, 32070752498/196703507, ... queen graph A000000/A000000 0, 12, 5010945279/34516916, ... rook graph A000000/A000000 0, 28/3, 1053/10, 2688/5, 38750/21, 34830/7, 45619/4, ... rook complement graph A000000/A000000 0, X, 1053/10, 42048/55, 185000/57, 2081700/203, ... Sierpiński carpet graph A000000/A000000 4264/105, ... Sierpiński gasket graph A000000/A000000 9/2, 6417/220, 484320175141452684064149/1906583190897703991200, ... Sierpiński tetrahedron graph A000000/A000000 12, 14752/105, ... star graph A000217 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... sun graph A000000/A000000 X, X, 6417/220, 918734/14839, 1144279285/10364816, ... sunlet graph A000000/A000000 X, X, 723/40, 2734/77, 38765/693, 110163/1309, 1115891/9690, ... tetrahedral Johnson graph A000000/A000000 X, X, X, X, X, 1061940/817, 148345050/24973, 8420160/407, ... torus grid graph A000000/A000000 X, X, X, 1053/10, 2766448/6405, 701375/612, ... transposition graph A000000/A000000 0, 1, 171/5, 2673408/1771, 1257873200406016/14301920633, ... triangular graph A000000/A000000 X, 0, 9/2, 204/5, 1300/7, 8325/14, 6125/4, 10192/3, ... triangular grid graph A000000/A000000 9/2, 6417/220, 213639983/2017730, ... web graph A000000/A000000 X, X, 7639771/124982, 27888076/226765, ... wheel graph A000000/A000000 X, X, X, 12, 309/14, 407/12, 3151/66, 437059/6864, 3678353/44979, ... white bishop graph A000000/A000000 X, 1, 28/3, 242259454379/3706922420, 677409758654/3149811665, ...

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.

Wiener Sum Index

## Cite this as:

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