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Wiener Sum Index

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The Wiener sum index WS is a topological index defined for a finite connected graph on n>=1 vertices by

WS=sum_(1<=i<j<=n)((d)_(ij))/((Omega)_(ij)),

where (d)_(ij) and (Omega)_(ij) are entries of the graph distance matrix and resistance distance matrix, respectively. The singleton graph has WS=0 by the empty sum convention.

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 Wiener sum index for various special classes of graphs. The index of each value list in the table starts at 1. An X indicates that the family in that row is not defined or is disconnected for the corresponding parameter value.

graph classOEISWS(G_1), WS(G_2), ...
Andrásfai graph1, 175/12, 84959/1353, 4384933627811/26124236280, ...
antiprism graphX, X, 204/5, 532238/6745, 47877676/368433, 906597918/4666585, ...
Apollonian network12, 2904467/50616, ...
black bishop graph n×n0, 1, 15, 242259454379/3706922420, ...
cocktail party graphX, 28/3, 204/5, 744/7, 1960/9, 4260/11, 8148/13, 14224/15, ...
complete bipartite graph K_(n,n)1, 28/3, 171/5, 592/7, 1525/9, 3276/11, 6223/13, 10816/15, ...
complete tripartite graph K_(n,n,n)9/2, 204/5, 1161/8, 3888/11, 9825/14, 20844/17, 39249/20, ...
complete graph K_n0, 1, 9/2, 12, 25, 45, 147/2, 112, 162, 225, ...
2n-crossed prism graphX, 2344/35, 295518737/1855364, 1143904996/3804075, ...
crown graph K_(n,n)-MX, X, 111/5, 2344/35, 9145/63, 2924/11, 62629/143, 130768/195, ...
cube-connected cycle graphX, X, 2373085829482/3510505999, ...
cycle graph C_nX, X, 9/2, 28/3, 175/12, 111/5, 1813/60, 4264/105, 14391/280, ...
Fibonacci cube graph1, 3, 593/42, 65264215739843/1291887161565, ...
folded cube graph, dimension n+11, 12, 592/7, 9920/21, 3237632/1147, 4195744/279, ...
gear graphX, X, 18905/462, 107212/1479, 211893947/1898424, ...
grid graph P_n square P_n0, 28/3, 107212/1479, ...
grid graph P_n square P_n square P_n0, 2344/35, ...
halved cube graph0, 1, 12, 744/7, 2432/3, 228431520/40703, 253056/7, ...
Hanoi graph9/2, 17830832037/286489840, ...
hypercube graph Q_n1, 28/3, 2344/35, 2766448/6405, 292437526/112065, ...
Keller graphX, 9920/21, ...
king graph n×n0, 12, 65906799847382/641818867935, ...
knight graph n×n0, X, X, 3736238283778148/12685333796135, ...
Menger sponge graph647093662/1662595, ...
Möbius ladderX, X, 171/5, 84959/1353, 859447105/8006068, ...
Mycielski graph0, 1, 175/12, 18680381340574/124654958285, ...
odd graph O_n0, 9/2, 100, 9678725/4301, 43454290887/873115, ...
pan graphX, 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_n0, 1, 111/5, 2030300070/2679523, ...
prism graph Y_nX, X, 1435/44, 2344/35, 102133740/954767, 32070752498/196703507, ...
queen graph n×n0, 12, 5010945279/34516916, ...
rook graph K_n square K_n0, 28/3, 1053/10, 2688/5, 38750/21, 34830/7, 45619/4, ...
rook complement graph K_n square K_n^_0, X, 1053/10, 42048/55, 185000/57, 2081700/203, ...
Sierpiński carpet graph4264/105, ...
Sierpiński gasket graph9/2, 6417/220, 484320175141452684064149/1906583190897703991200, ...
Sierpiński tetrahedron graph12, 14752/105, ...
star graph S_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ...
sun graphX, X, 6417/220, 918734/14839, 1144279285/10364816, ...
sunlet graph C_n circledot K_1X, X, 723/40, 2734/77, 38765/693, 110163/1309, 1115891/9690, ...
tetrahedral Johnson graphX, X, X, X, X, 1061940/817, 148345050/24973, 8420160/407, ...
torus grid graph C_n square C_nX, X, 1053/10, 2766448/6405, 701375/612, ...
transposition graph0, 1, 171/5, 2673408/1771, 1257873200406016/14301920633, ...
triangular graphX, 0, 9/2, 204/5, 1300/7, 8325/14, 6125/4, 10192/3, ...
triangular grid graph9/2, 6417/220, 213639983/2017730, ...
web graphX, X, 7639771/124982, 27888076/226765, ...
wheel graph W_nX, X, X, 12, 309/14, 407/12, 3151/66, 437059/6864, 3678353/44979, ...
white bishop graph n×nX, 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 and |_x_| is the floor function. The parameters are positive integers, with additional restrictions indicated in the table.

graph classWS(n)
cocktail party graph n>=2(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_(n,n)-M n>=3(n(6n^4-17n^3+6n^2+12n-6))/((2n-3)(2n-1))
cycle graph C_n n>=3n^2(H_(2|_n/2_|)-H_(|_n/2_|))
path graph P_n1/2n(n-1)
rook complement graph K_n square K_n^_ n>=3((n-1)^2n^4(n^3+2n^2-4n-7))/(4(n+1)(n^2-n-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-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

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Application." 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, 1999.Randić, M. "On Characterization of Cyclic Structures." J. Chem. Inf. Comput. Sci. 37, 1063-1071, 1997.Sloane, N. J. A. Sequence A000217 in "The On-Line Encyclopedia of Integer Sequences."

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Wiener Sum Index

Cite this as:

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

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