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علماء الرياضيات | Wiener Index |
 1785
 04:25 مساءً
date: 8-4-2022 |
Author : Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N 
Book or Source : "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90 
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The Wiener index 
, denoted 
(Wiener 1947) and also known as the "path number" or Wiener number (Plavšić et al. 1993), is a graph index defined for a graph on 
nodes by
 |
where 
is the graph 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 Wiener index is not very discriminant. In fact, the paw graph and square graph on four nodes are already indistinguishable using the Wiener index (both have value 8). The numbers of non-Wiener-unique connected graphs on 
, 2, ... nodes given by 0, 0, 0, 2, 16, 108, 847, 11110, 261072, ... (OEIS A193217).
Precomputed values for many graphs are implemented in the Wolfram Language as GraphData[g, "WienerIndex"].
The following table summarizes values of the Wiener index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A292018 | 1, 15, 44, 88, 147, 221, 310, 414, ... |
antelope graph  |
A292039 | 0,  ,  ,  ,  ,  ,  , 11548, 16660, ... |
| antiprism graph | A002411 | X, X, 18, 40, 75, 126, 196, 288, ... |
| Apollonian network | A289022 | 6, 27, 204, 1941, 19572, 198567, ... |
black bishop graph  |
A292051 | 0, 1, 14, 42, 124, 251, 506, 852, 1432, 2165, ... |
| cocktail party graph | A001105 |  , 8, 18, 32, 50, 72, 98, 128, 162, ... |
complete bipartite graph  |
A000567 | 1, 1, 5, 73, 2069, 95401, 6487445, ... |
complete tripartite graph  |
A094159 | 1, 11, 1243, 490043, 463370491, ... |
complete graph  |
A000217 | 0, 1, 3, 6, 10, 15, 21, 28, 36, ... |
 -crossed prism graph |
A292022 | X, 48, 132, 288, 540, 912, 1428, ... |
crown graph  |
A033428 | X, X, 27, 48, 75, 108, 147, 192, 243, ... |
| cube-connected cycle graph | A292028 | X, X, 888, 9472, 76336, 559584, 3594952, ... |
cycle graph  |
A034828 | X, X, 3, 8, 15, 27, 42, 64, 90, ... |
| Fibonacci cube graph | A238419 | 1, 4, 16, 54, 176, 548, 1667, 4968,, ... |
fiveleaper graph  |
A292040 | 0,  ,  ,  ,  ,  ,  , 6364, 9888, 15216, ... |
| folded cube graph | A292029 | X, 1, 6, 40, 200, 1056, 4928, 23808, ... |
| gear graph | A049598 | X, X, 36, 72, 120, 180, 252, 336, 432, ... |
grid graph  |
A143945 | 0, 8, 72, 320, 1000, 2520, 5488, 10752, ... |
grid graph  |
A292045 | 0, 48, 972, 7680, 37500, 136080, 403368, ... |
| halved cube graph | A292044 | 0, 1, 6, 32, 160, 768, 3584, 16384, ... |
| Hanoi graph | A290004 | 3, 72, 1419, 26580, 487839, 8867088, ... |
hypercube graph  |
A002697 | 1, 8, 48, 256, 1280, 6144, 28672, ... |
| Keller graph | A292056 |  , 200, 2944, 43392, 650240, 9889792, ... |
king graph  |
A292053 | 0, 6, 52, 228, 708, 1778, 3864, 7560, ... |
knight graph  |
A292054 | 0,  ,  , 288, 708, 1580, 3144, 5804, 9996, ... |
| Menger sponge graph | A292036 | 612, 794976, 954380016, ... |
| Möbius ladder | A180857 | X, X, 21, 44, 85, 138, 217, 312, 441, ... |
| Mycielski graph | A292055 | 0, 1, 15, 90, 435, 1926, 8175, 33930, ... |
odd graph  |
A136328 | 0, 3, 75, 1435, 25515, 436821, ... |
| pan graph | A180861 | 8, 16, 26, 42, 61, 88, 119, 160, 206, 264, ... |
path graph  |
A000292 | 0, 1, 4, 10, 20, 35, 56, 84, 120, ... |
permutation star graph  |
A284039 | 0, 1, 27, 744, 26520, 1239840, ... |
prism graph  |
A138179 | X, X, 21, 48, 85, 144, 217, 320, 441, ... |
queen graph  |
A292057 | 0, 6, 44, 164, 440, 970, 1876, 3304, 5424, ... |
rook graph  |
A085537 | X, 8, 54, 192, 500, 1080, 2058, 3584, 5832, ... |
rook complement graph  |
A292058 | 0,  , 54, 168, 400, 810, 1470, 2464, ... |
| Sierpiński carpet graph | A292025 | 64, 13224, 2535136, 485339728, ... |
| Sierpiński sieve graph | A290129 | 3, 21, 246, 3765, 64032, 1130463, 20215254, ... |
| Sierpiński tetrahedron graph | A292026 | 6, 66, 1476, 42984, 1343568, 42744480, ... |
star graph  |
A000290 | 0, 1, 4, 9, 16, 25, 36, 49, 64, ... |
| sun graph | A180863 | X, X, 21, 44, 75, 114, 161, 216, 279, 350, ... |
sunlet graph  |
A180574 | X, X, 27, 60, 105, 174, 259, 376, 513, 690, ... |
| tetrahedral graph | A292061 | X, X, X, X, X, 300, 1050, 2940, 7056, 15120, ... |
torus grid graph  |
A122657 | 54, 256, 750, 1944, 4116, 8192, 14580, 25000, ... |
| transposition graph | A292062 | 0, 1, 21, 552, 19560, 920160, 55974240, ... |
| triangular graph | A006011 | 0, 3, 18, 60, 150, 315, 588, 1008, 1620, ... |
| triangular grid graph | A112851 | 3, 21, 81, 231, 546, 1134, 2142, 3762, 6237, ... |
| web graph | A180576 | X, X, 69, 148, 255, 417, 616, 888, 1206, 1615, ... |
wheel graph  |
A002378 | X, X, X, X, 12, 20, 30, 42, 56, 72, ... |
white bishop graph  |
A292059 | X, 1, 8, 42, 104, 251, 464, 852, 1360, 2165, ... |
Closed forms are summarized in the following table. The cycle graph was considered by Plavšić et al. (1993) and Babić et al. (2002) and the path graph by Plavšić et al. (1993).
| Andrásfai graph |  |
| antiprism graph |  |
cocktail party graph  |
 |
complete graph  |
 |
| crossed prism graph |  |
crown graph  |
 |
cycle graph  ,  |
 |
| gear graph |  |
grid graph  |
 |
grid graph  |
 |
| halved cube graph |  |
hypercube graph  |
 |
| Möbius ladder | ![1/4n[2n(n+2)-3-(-1)^n]](https://mathworld.wolfram.com/images/equations/WienerIndex/Inline71.svg) |
Mycielski graph  |
 |
path graph  |
 |
rook graph  |
 |
star graph  |
 |
| sun graph |  |
sunlet graph  |
![1/4n[(-1)^n+2n(n+4)-5]](https://mathworld.wolfram.com/images/equations/WienerIndex/Inline82.svg) |
| triangular graph |  |
wheel graph  |
 |
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.
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 26 and 108-109, 1999.
Hosoya, H. "Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons." Bull. Chem. Soc. Japan 44, 2322-2239, 1971.
Plavšić, D.; Nikolić, S.; Trinajstić, N.; and Mihalić, Z. "On the Harary Index for the Characterization of Chemical Graphs." J. Math. Chem. 12, 235-250, 1993.
Sloane, N. J. A. Sequence OEIS A193217 in "The On-Line Encyclopedia of Integer Sequences."Wiener, H. J. "Structural Determination of Paraffin Boiling Points." J. Amer. Chem. Soc. 69, 17-20, 1947.
Wiener, H. "Influence of Interatomic Forces on Paraffin Properties." J. Chem. Phys. 15, 766, 1947.Wiener, H. "Vapor Pressure-Temperature Relationships Among the Branched Paraffin Hydrocarbons." J. Phys. Chem. 52, 425-430, 1948.
Wiener, H. "Relation of the Physical Properties of the Isomeric Alkanes to Molecular Structure. Surface Tension, Specific Dispersion, and Critical Solution Temperature in Aniline." J. Phys. Chem. 52, 1082-1089, 1948.
Zerovnik, J. "Szeged Index of Symmetric Graphs." J. Chem. Inf. Comput. Sci. 39, 77-80, 1999.
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