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Wiener Sum Index

المؤلف:  Devillers, J. and Balaban, A. T

المصدر:  Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, 1999.

الجزء والصفحة:  ...

8-4-2022

2118

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 sieve 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 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.

graph class
cocktail party graph
complete bipartite graph
complete bipartite graph
complete graph
complete tripartite graph
crown graph
cycle graph
path graph
rook complement graph
rook graph
star graph
triangular graph

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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.