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Harary Index
المؤلف:
Devillers, J. and Balaban, A. T
المصدر:
Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach
الجزء والصفحة:
...
7-4-2022
3018
+
-
20
Harary Index
The Harary index of a graph 
on 
vertices was defined by Plavšić et al. (1993) as
 |
(1) |
where
|
(2) |
is the reciprocal of the graph distance matrix 
(Plavšić et al. 1993; Devillers and Balaban, p. 80, 2000).
Some care is needed, since while some authors include the leading factor of 1/2 (e.g., Plavšić et al. 1993, Mercader et al. 2001), others omit it (e.g., Devillers and Balaban 1999, pp. 111 and 202).
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 Harary index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A000000/A000000 | 1, 15/2, 20, 77/2, 63, 187/2, 130, 345/2, 221, ... |
| antiprism graph | A000000/A000000 | X, X, 27/2, 22, 95/3, 42, 637/12, 194/3, 384/5, ... |
| Apollonian network | A000000/A000000 | 6, 18, 80, 470, 3248, 122106/5, 3394391/20, 6406407/20, ... |
bishop graph  |
A296197 | 0, 2, 13, 42, 102, 208, 379, 636, 1004, 1510, ... |
black bishop graph  |
A296198 | 0, 1, 8, 21, 55, 104, 197, 318, 514, 755, ... |
cocktail party graph  |
A000000/A000000 | 0, 5, 27/2, 26, 85, 126, 175, 232, 297, 370, ... |
complete bipartite graph  |
A000326 | 2, 5, 12, 44, 70, 102, 140, 184, ... |
complete graph  |
A000217 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... |
complete tripartite graph  |
A000000/A000000 | 3, 27/2, 63, 114, 180, 261, ... |
 -crossed prism graph |
A000000/A000000 | 58/3, 39, 368/3, 514/3, 1116/5, 4166/15, 35128/105, ... |
| crown graph | A000000/A000000 | X, X, 10, 58/3, 95/3, 47, 196/3, 260/3, 111, 415/3, ... |
| cube-connected cycle graph | A000000/A000000 | X, X, 556/5, 57376/105, 162634/63, 34149904/3003, ... |
cycle graph  |
A160046/A160047 | X, X, 3, 5, 15/2, 10, 77/6, 47/3, 75/4, 131/6, ... |
| Fibonacci cube graph | A000000/A000000 | 1, 5/2, 22/3, 71/4, 216/5, 1219/12, 25033/105, ... |
| folded cube graph | A000000/A000000 | X, 1, 6, 22, 80, 808/3, 2800/3, 9488/3, 11072, ... |
| gear graph | A000000/A000000 | X, X, 29/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ... |
grid graph  |
A296191/A296192 | 0, 5, 133/6, 293/5, 3399/28, 137111/630, 140351/396, ... |
grid graph  |
A000000/A000000 | 0, 58/3, 2402/15, 30617/45, 7168769/3465, ... |
| halved cube graph | A290347/A290348 | 0, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, ... |
| Hanoi graph | A000000/A000000 | 3, 22, 4276/35, 1835837/3003, 175359949924361/60168147039, ... |
| helm graph | A000000/A000000 | 29/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ... |
hypercube graph  |
A290343/A290344 | 1, 5, 58/3, 206/3, 3548/15, 12136/15, 291824/105, ... |
Keller graph  |
A296189 | 0, 80, 1552, 27264, 460544, 7634944, ... |
king graph  |
A144945 | 0, 6, 28, 76, 160, 290, 476, 728, 1056, 1470, ... |
knight graph  |
A000000/A000000 | 0, 0, 47/3, 309/5, 150, 1769/6, 7724/15, 24733/30, ... |
| Menger sponge graph | A000000/A000000 | 1147/15, 207460203161/19684665, ... |
| Möbius ladder | A000000/A000000 | X, X, 12, 20, 85/3, 38, 287/6, 176/3, 348/5, 244/3, ... |
| Mycielski graph | A296193/A000000 | 0, 1, 15/2, 75/2, 162, 1317/2, 2610, 20505/2, 40212, ... |
odd graph  |
A000000 | 0, 3, 30, 280, 2730, 57057/2, 635635/2, ... |
| pan graph | A000000/A000000 | X, X, 5, 22/3, 61/6, 155/12, 16, 571/30, 1339/60, ... |
path graph  |
A160048/A160049 | 0, 2, 5, 26/3, 77/6, 87/5, 223/10, 962/35, ... |
permutation star graph  |
A296190/A296057 | 0, 1, 10, 123, 2202, 59040, 2287680, 121394000, ... |
prism graph  |
A000000/A000000 | X, X, 12, 58/3, 85/3, 75/2, 287/6, 874/15, ... |
queen graph  |
A296196 | 0, 6, 32, 98, 230, 460, 826, 1372, 2148, 3210, ... |
rook complement graph  |
A092364 | 0, 2, 27, 96, 250, 540, 1029, 1792, 2916, 4500, ... |
rook graph  |
A085740 | X, 5, 54, 168, 400, 810, 1470, 2464, 3888, 5850, ... |
| Sierpiński carpet graph | A000000/A000000 | 47/3, 23255059/51480, ... |
| Sierpiński sieve graph | A000000/A000000 | 3, 12, 227/4, 5553/20, 161390213/120120, ... |
| Sierpiński tetrahedron graph | A000000/A000000 | 6, 69/2, 1055/4, 599803/280, 279423163/16016, ... |
star graph  |
A160050/A130658 | 0, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ... |
| sun graph | A000000/A000000 | X, X, 10, 97/6, 95/4, 158/5, 2429/60, 743/15, ... |
sunlet graph  |
A000000/A000000 | X, X, 10, 97/3, 95/2, 316/5, 2429/30, 1486/15, 594/5, ... |
| tetrahedral graph | A000000/A000000 | X, X, 415/3, 2345/6, 2800/3, 1981, 3850, 6985, 11990, ... |
torus grid graph  |
A000000/A000000 | X, X, 27, 206/3, 875/6, 1287/5, 12691/30, 66964/105, ... |
| transposition graph | A296194 | 0, 1, 12, 162, 3010, 81000, 3105396, 162469104, ... |
| triangular graph | A000000/A000000 | X, 0, 3, 27/2, 75/2, 165/2, 315/2, 273, 441, 675, 990, ... |
| triangular grid graph | A027480 | 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, ... |
| web graph | A000000/A000000 | X, X, 45/2, 217/6, 635/12, 703/10, 1799/20, 110, ... |
wheel graph  |
A000000/A000000 | 6, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51, 117/2, ... |
white bishop graph  |
A296200 | 1, 5, 21, 47, 104, 182, 318, 490, 755, ... |
Closed forms for some special classes are summarized in the following table. Here, 
is a harmonic number, 
is a Catalan number, 
is a Lerch transcendent, 
is a generalized hypergeometric function, and 
is a Stirling number of the first kind.
| graph | Harary index |
| Andrásfai graph |  |
| antiprism graph | |
bishop graph  |
![1/(48)[6n^4+16n^3-36n^2+8n+3-3(-1)^n]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline40.svg) |
black bishop graph  |
![1/(96)[6n^4+16n^3-30n^2+20n-15-3(-1)^n(2n^2+4n-5)]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline42.svg) |
cocktail party graph  |
 |
complete bipartite graph  |
 |
complete bipartite graph  |
 |
complete graph  |
 |
complete tripartite graph  |
 |
 -crossed prism graph |
 |
crown graph  |
 |
cycle graph  |
 |
empty graph  |
0 |
gear graph  |
 |
grid graph  |
![1/3n(2n+1)[(2n-1)(H_(n-3/2)-H_n+2ln2)-n+2]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline63.svg) |
![=1/3n(2n+1)[(2n-1)(dlnC_(n+1))/(dn)-n+2]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline64.svg) |
|
halved cube graph  |
![-2^(n-1)H_n-2^(2n-1)R[Phi(2,1,n+1)]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline66.svg) |
| helm graph |  |
hypercube graph  |
 |
| Keller graph | |
king graph  |
 |
Möbius ladder  |
|
Mycielski graph  |
|
| pan graph | |
path graph  |
 |
prism graph  |
|
queen graph  |
 |
rook complement graph  |
 |
rook graph  |
 |
star graph  |
 |
| sun graph |  |
sunlet graph  |
|
| tetrahedral graph |  |
torus grid graph  |
|
| transposition graph |  |
| triangular graph |  |
| triangular grid graph |  |
| web graph | |
wheel graph  |
 |
white bishop graph  |
![1/(96)[6n^4+16n^3-42n^2-4n+21+3(-1)^n(2n^2+4n-7)]](https://mathworld.wolfram.com/images/equations/HararyIndex/Inline103.svg) |
REFERENCES
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 40, 111, 202, and 227, 1999.
Diudea, M. V.; Ivanciuc, T.; Nikolić, S.; and Trinajstić, N. "Matrices of Reciprocal Distance, Polynomials and Derived Numbers." MATCH (Commun. Math. Comput. Chem.) 35, 41-64, 1997.
Ivanciuc, O.; Balaban, T.-S.; and Balaban, A. T. "Design of Topological Indices. Part 4. Reciprocal Distance Matrix, Related Local Vertex Invariants and Topological Indices." J. Math. Chem. 12, 309-318, 1993.
Mercader, E.; Castro, E. A.; and Toropov, A. A. "Maximum Topological Distances Based Indices as Molecular Descriptors for QSPR. 4. Modeling the Enthalpy of Formation of Hydrocarbons from Elements." Int. J. Mol. Sci. 2, 121-132, 2001.
Mihalić, Z. and Trinajstić, N. "A Graph Theoretical Approach to Structure-Property Relationships." J. Chem. Educ. 69, 701-712, 1992.
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. Sequences A000217, A160046, A160047, A160048, A160049, A160050, A290343, A290344, A290347, and A290348 in "The On-Line Encyclopedia of Integer Sequences."
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