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علماء الرياضيات | Wiener Sum Index |
 1506
 04:32 مساءً
date: 8-4-2022 |
Author : Devillers, J. and Balaban, A. T 
Book or Source : Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, 1999. 
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Date: 15-5-2022
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Date: 24-2-2022
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Date: 3-8-2016
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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  |
![(mn[m^2+n^2+4mn-3(m+n)+2])/(2(m+n-1))](https://mathworld.wolfram.com/images/equations/WienerSumIndex/Inline35.svg) |
complete bipartite graph  |
 |
complete graph  |
 |
complete tripartite graph  |
 |
crown graph  |
 |
cycle graph  |
 |
path graph  |
 |
rook complement graph  |
 |
rook graph  |
 |
star graph  |
 |
| triangular graph |  |
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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