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Wiener Sum Index

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

WS=1/2sum_(i=1)^nsum_(j=1)^n((d)_(ij))/((Omega)_(ij)),

where (d)_(ij) is the graph distance matrix and (Omega)_(ij) 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 classOEISWS(G_1), WS(G_2), ...
Andrásfai graphA000000/A0000001, 175/12, 84959/1353, 4384933627811/26124236280, ...
antiprism graphA000000/A000000X, X, 204/5, 532238/6745, 47877676/368433, 906597918/4666585, ...
Apollonian networkA000000/A00000012, 2904467/50616, ...
black bishop graph n×nA000000/A0000000, 1, 15, 242259454379/3706922420, ...
cocktail party graphA000000/A000000X, 28/3, 204/5, 744/7, 1960/9, 4260/11, 8148/13, 14224/15, ...
complete bipartite graph K_(n,n)A000000/A0000001, 28/3, 171/5, 592/7, 1525/9, 3276/11, 6223/13, 10816/15, ...
complete tripartite graph K_(n,n,n)A000000/A0000009/2, 204/5, 1161/8, 3888/11, 9825/14, 20844/17, 39249/20, ...
complete graph K_nA000000/A0000000, 1, 9/2, 12, 25, 45, 147/2, 112, 162, 225, ...
2n-crossed prism graphA000000/A000000X, 2344/35, 295518737/1855364, 1143904996/3804075, ...
crown graph K_2 square K_n^_A000000/A000000X, X, 111/5, 2344/35, 9145/63, 2924/11, 62629/143, 130768/195, ...
cube-connected cycle graphA000000/A000000X, X, 2373085829482/3510505999, ...
cycle graph C_nA000000/A000000X, X, 9/2, 28/3, 175/12, 111/5, 1813/60, 4264/105, 14391/280, ...
Fibonacci cube graphA000000/A0000001, 3, 593/42, 65264215739843/1291887161565, ...
folded cube graphA000000/A0000001, 12, 592/7, 9920/21, 3237632/1147, 4195744/279, ...
gear graphA000000/A000000X, X, 18905/462, 107212/1479, 211893947/1898424, ...
grid graph P_n square P_nA000000/A0000000, 28/3, 107212/1479, ...
grid graph P_n square P_n square P_nA000000/A0000000, 2344/35, ...
halved cube graphA000000/A0000000, 1, 12, 744/7, 2432/3, 228431520/40703, 253056/7, ...
Hanoi graphA000000/A0000009/2, 17830832037/286489840, ...
hypercube graph Q_nA000000/A0000001, 28/3, 2344/35, 2766448/6405, 292437526/112065, ...
Keller graphA000000/A000000X, 9920/21, ...
king graph n×nA000000/A0000000, 12, 65906799847382/641818867935, ...
knight graph n×nA000000/A0000000, X, X, 3736238283778148/12685333796135, ...
Menger sponge graphA000000/A000000647093662/1662595, ...
Möbius ladderA000000/A000000X, X, 171/5, 84959/1353, 859447105/8006068, ...
Mycielski graphA000000/A0000000, 1, 175/12, 18680381340574/124654958285, ...
odd graph O_nA000000/A0000000, 9/2, 100, 9678725/4301, 43454290887/873115, ...
pan graphA000000/A000000X, X, 79/10, 593/42, 8131/396, 11378/385, 9772367/251940, ...
path graph P_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ...
permutation star graph PS_nA000000/A0000000, 1, 111/5, 2030300070/2679523, ...
prism graph Y_nA000000/A000000X, X, 1435/44, 2344/35, 102133740/954767, 32070752498/196703507, ...
queen graph n×nA000000/A0000000, 12, 5010945279/34516916, ...
rook graph K_n square K_nA000000/A0000000, 28/3, 1053/10, 2688/5, 38750/21, 34830/7, 45619/4, ...
rook complement graph K_n square K_n^_A000000/A0000000, X, 1053/10, 42048/55, 185000/57, 2081700/203, ...
Sierpiński carpet graphA000000/A0000004264/105, ...
Sierpiński gasket graphA000000/A0000009/2, 6417/220, 484320175141452684064149/1906583190897703991200, ...
Sierpiński tetrahedron graphA000000/A00000012, 14752/105, ...
star graph S_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ...
sun graphA000000/A000000X, X, 6417/220, 918734/14839, 1144279285/10364816, ...
sunlet graph C_n circledot K_1A000000/A000000X, X, 723/40, 2734/77, 38765/693, 110163/1309, 1115891/9690, ...
tetrahedral Johnson graphA000000/A000000X, X, X, X, X, 1061940/817, 148345050/24973, 8420160/407, ...
torus grid graph C_n square C_nA000000/A000000X, X, X, 1053/10, 2766448/6405, 701375/612, ...
transposition graphA000000/A0000000, 1, 171/5, 2673408/1771, 1257873200406016/14301920633, ...
triangular graphA000000/A000000X, 0, 9/2, 204/5, 1300/7, 8325/14, 6125/4, 10192/3, ...
triangular grid graphA000000/A0000009/2, 6417/220, 213639983/2017730, ...
web graphA000000/A000000X, X, 7639771/124982, 27888076/226765, ...
wheel graph W_nA000000/A000000X, X, X, 12, 309/14, 407/12, 3151/66, 437059/6864, 3678353/44979, ...
white bishop graph n×nA000000/A000000X, 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 H_n denotes a harmonic number.

graph classWS(n)
cocktail party graph(2(n-1)n(2n^2-1))/(2n-1)
complete bipartite graph K_(m,n)(mn[m^2+n^2+4mn-3(m+n)+2])/(2(m+n-1))
complete bipartite graph K_(n,n)(n^2(3n^2-3n+1))/(2n-1)
complete graph K_n1/4n^2(n-1)
complete tripartite graph K_(n,n,n)(3n^2(6n^2-4n+1))/(3n-1)
crown graph K_2 square K_n^_(n(6n^4-17n^3+6n^2+12n-6))/((2n-3)(2n-1))
cycle graph C_nn^2(H_(2|_n/2_|)-H_(|_n/2_|))
path graph P_n1/2n(n-1)
rook complement graph K_n square K_n^_((n-1)^2n^4(n^3+2n^2-4n-7))/(4(n+1)(n^2-n1-1))
rook graph K_n square K_n((n-1)n^4(n^2+n+1))/((n+1)(n+2))
star graph S_n1/2n(n-1)
triangular graph((n-2)(n-1)^2n^2(n^2+1))/(4(n+1)(n+2))

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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