Wiener Index

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 ,	{1/8n^3 for n even; 1/8(n-1)n(n+1) for n odd" src="https://mathworld.wolfram.com/images/equations/WienerIndex/Inline62.svg" style="height:76px; width:314px" />
gear graph
grid graph
grid graph
halved cube graph	{0 for n=1; n·2^(2n-5) for n>=2" src="https://mathworld.wolfram.com/images/equations/WienerIndex/Inline68.svg" style="height:68px; width:205px" />
hypercube graph
Möbius ladder
Mycielski graph
path graph
rook graph
star graph
sun graph
sunlet graph
triangular graph
wheel graph

---

REFERENCES

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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