TOPICS
Search

Harary Index

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The Harary index of a graph G on n vertices was defined by Plavšić et al. (1993) as

H(G)=1/2sum_(i=1)^nsum_(j=1)^n(RD)_(ij),
(1)

where

(RD)_(ij)={D_(ij)^(-1) if i!=j; 0 if i=j
(2)

is the reciprocal of the graph distance matrix D (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 classOEISH(G_1), H(G_2), ...
Andrásfai graphA000000/A0000001, 15/2, 20, 77/2, 63, 187/2, 130, 345/2, 221, ...
antiprism graphA000000/A000000X, X, 27/2, 22, 95/3, 42, 637/12, 194/3, 384/5, ...
Apollonian networkA000000/A0000006, 18, 80, 470, 3248, 122106/5, 3394391/20, 6406407/20, ...
bishop graph B_(n,n)A2961970, 2, 13, 42, 102, 208, 379, 636, 1004, 1510, ...
black bishop graph BB_(n,n)A2961980, 1, 8, 21, 55, 104, 197, 318, 514, 755, ...
cocktail party graph K_(n×2)A000000/A0000000, 5, 27/2, 26, 85, 126, 175, 232, 297, 370, ...
complete bipartite graph K_(n,n)A0003262, 5, 12, 44, 70, 102, 140, 184, ...
complete graph K_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ...
complete tripartite graph K_(n,n,n)A000000/A0000003, 27/2, 63, 114, 180, 261, ...
2n-crossed prism graphA000000/A00000058/3, 39, 368/3, 514/3, 1116/5, 4166/15, 35128/105, ...
crown graphA000000/A000000X, X, 10, 58/3, 95/3, 47, 196/3, 260/3, 111, 415/3, ...
cube-connected cycle graphA000000/A000000X, X, 556/5, 57376/105, 162634/63, 34149904/3003, ...
cycle graph C_nA160046/A160047X, X, 3, 5, 15/2, 10, 77/6, 47/3, 75/4, 131/6, ...
Fibonacci cube graphA000000/A0000001, 5/2, 22/3, 71/4, 216/5, 1219/12, 25033/105, ...
folded cube graphA000000/A000000X, 1, 6, 22, 80, 808/3, 2800/3, 9488/3, 11072, ...
gear graphA000000/A000000X, X, 29/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ...
grid graph P_n square P_nA296191/A2961920, 5, 133/6, 293/5, 3399/28, 137111/630, 140351/396, ...
grid graph P_n square P_n square P_nA000000/A0000000, 58/3, 2402/15, 30617/45, 7168769/3465, ...
halved cube graphA290347/A2903480, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, ...
Hanoi graphA000000/A0000003, 22, 4276/35, 1835837/3003, 175359949924361/60168147039, ...
helm graphA000000/A00000029/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ...
hypercube graph Q_nA290343/A2903441, 5, 58/3, 206/3, 3548/15, 12136/15, 291824/105, ...
Keller graph K_nA2961890, 80, 1552, 27264, 460544, 7634944, ...
king graph Ki_(n,n)A1449450, 6, 28, 76, 160, 290, 476, 728, 1056, 1470, ...
knight graph Kn_(n,n)A000000/A0000000, 0, 47/3, 309/5, 150, 1769/6, 7724/15, 24733/30, ...
Menger sponge graphA000000/A0000001147/15, 207460203161/19684665, ...
Möbius ladderA000000/A000000X, X, 12, 20, 85/3, 38, 287/6, 176/3, 348/5, 244/3, ...
Mycielski graphA296193/A0000000, 1, 15/2, 75/2, 162, 1317/2, 2610, 20505/2, 40212, ...
odd graph O_nA0000000, 3, 30, 280, 2730, 57057/2, 635635/2, ...
pan graphA000000/A000000X, X, 5, 22/3, 61/6, 155/12, 16, 571/30, 1339/60, ...
path graph P_nA160048/A1600490, 2, 5, 26/3, 77/6, 87/5, 223/10, 962/35, ...
permutation star graph PS_nA296190/A2960570, 1, 10, 123, 2202, 59040, 2287680, 121394000, ...
prism graph Y_nA000000/A000000X, X, 12, 58/3, 85/3, 75/2, 287/6, 874/15, ...
queen graph Q_(n,n)A2961960, 6, 32, 98, 230, 460, 826, 1372, 2148, 3210, ...
rook complement graph K_n square K_n^_A0923640, 2, 27, 96, 250, 540, 1029, 1792, 2916, 4500, ...
rook graph K_n square K_nA085740X, 5, 54, 168, 400, 810, 1470, 2464, 3888, 5850, ...
Sierpiński carpet graphA000000/A00000047/3, 23255059/51480, ...
Sierpiński gasket graphA000000/A0000003, 12, 227/4, 5553/20, 161390213/120120, ...
Sierpiński tetrahedron graphA000000/A0000006, 69/2, 1055/4, 599803/280, 279423163/16016, ...
star graph S_nA160050/A1306580, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ...
sun graphA000000/A000000X, X, 10, 97/6, 95/4, 158/5, 2429/60, 743/15, ...
sunlet graph C_n circledot K_1A000000/A000000X, X, 10, 97/3, 95/2, 316/5, 2429/30, 1486/15, 594/5, ...
tetrahedral Johnson graphA000000/A000000X, X, 415/3, 2345/6, 2800/3, 1981, 3850, 6985, 11990, ...
torus grid graph C_n square C_nA000000/A000000X, X, 27, 206/3, 875/6, 1287/5, 12691/30, 66964/105, ...
transposition graphA2961940, 1, 12, 162, 3010, 81000, 3105396, 162469104, ...
triangular graphA000000/A000000X, 0, 3, 27/2, 75/2, 165/2, 315/2, 273, 441, 675, 990, ...
triangular grid graphA0274803, 12, 30, 60, 105, 168, 252, 360, 495, 660, ...
web graphA000000/A000000X, X, 45/2, 217/6, 635/12, 703/10, 1799/20, 110, ...
wheel graph W_nA000000/A0000006, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51, 117/2, ...
white bishop graph WB_(n,n)A2962001, 5, 21, 47, 104, 182, 318, 490, 755, ...

Closed forms for some special classes are summarized in the following table. Here, H_n is a harmonic number, C_n is a Catalan number, Phi(z,s,a) is a Lerch transcendent, _pF_q is a generalized hypergeometric function, and s(n,m) is a Stirling number of the first kind.

graphHarary index
Andrásfai graph1/2(3n-1)(2n-1)
antiprism graph{2(2nH_(n/2-1)+3) for n even; 2n(2H_((n-1)/2)+1/(n+1)) for n odd
bishop graph B_(n,n)1/(48)[6n^4+16n^3-36n^2+8n+3-3(-1)^n]
black bishop graph BB_(n,n)1/(96)[6n^4+16n^3-30n^2+20n-15-3(-1)^n(2n^2+4n-5)]
cocktail party graph K_(n×2)1/2n(4n-3)
complete bipartite graph K_(m,n)1/4(m^2+n^2-m-n)+mn
complete bipartite graph K_(n,n)1/2n(3n-1)
complete graph K_n1/2n(n-1)
complete tripartite graph K_(n,n,n)3/4n(5n-1)
2n-crossed prism graph(8H_(n-1)-1/3)n+4
crown graph K_2 square K_n^_1/6n(9n-7)
cycle graph C_n1/2(1+(-1)^n)+nH_(|_(n-1)/2_|)
empty graph K^__n0
gear graph G_n1/(24)n(17n+65)
grid graph P_n square P_n1/3n(2n+1)[(2n-1)(H_(n-3/2)-H_n+2ln2)-n+2]
=1/3n(2n+1)[(2n-1)(dlnC_(n+1))/(dn)-n+2]
halved cube graph 1/2Q_n-2^(n-1)H_n-2^(2n-1)R[Phi(2,1,n+1)]
helm graph1/(24)n(17n+65)
hypercube graph Q_n2^(n-1)n_3F_2(1,1,1-n;2,1;-1)
Keller graph{0 for n=1; 4^n[4^n-1/2(3^n+n+1)]/2 otherwise
king graph Ki_(n,n)1/3(n-1)n(5n-1)
Möbius ladder M_n{n(4H_(n/2)-1) for n even; n(4H_((n-1)/2)-(n-3)/(n+1)) for n odd
Mycielski graph M_n{0 for n=1; 1/(48)(36-452^n+283^(n-1)+274^(n-1)) otherwise
pan graph{(n+2)H_(n/2)+2/(n+1) for n even; (n+2)H_((n-1)/2)+4/(n+1)-1 for n odd
path graph P_nn(H_n-1)
prism graph Y_n{4(nH_(n/2-1)-1/(n+2)+2)-n for n even; n(H_((n-1)/2)-(n-3)/(n+1)) for n odd
queen graph Q_(n,n)1/(12)(n-1)n(3n^2+13n-2)
rook complement graph K_n square K_n^_1/2(n-1)n^3
rook graph K_n square K_n1/4(n-1)n^2(n+3)
star graph S_n1/4(n+2)(n-1)
sun graph1/6n(7n+3)
sunlet graph C_n circledot K_1{4nH_(n/2-1)-(5n)/2-8/(n+2)-4/(n+4)+12 for n even; 4nH_((n-1)/2)-(5n)/2-6/(n+1)-6/(n+3)+8 for n odd
tetrahedral Johnson graph1/(432)(n-3)(n-2)(n-1)n(2n^2+9n+40)
torus grid graph C_n square C_n{1/2n[4n^2(H_(n-1)-H_(n/2))+4n-3] for n even; 2n^3(H_(n-1)-H_((n-1)/2)) for n odd
transposition graph1/2sum_(k=1)^(n-1)((-1)^ks(n,n-k))/k
triangular graph1/(16)(n-2)(n-1)n(n+5)
triangular grid graph1/2n(n+1)(n+2)
web graph{9nH_(n/2-1)-6n-(20)/(n+2)-(12)/(n+4)+29 for n even; 9nH_((n-1)/2)-(2n(3n^2+2n-15))/((n+1)(n+3)) for n odd
wheel graph W_n1/4(n-1)(n+4)
white bishop graph WB_(n,n)1/(96)[6n^4+16n^3-42n^2-4n+21+3(-1)^n(2n^2+4n-7)]

See also

Graph Distance Matrix, Topological Index

Explore with Wolfram|Alpha

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."

Referenced on Wolfram|Alpha

Harary Index

Cite this as:

Weisstein, Eric W. "Harary Index." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HararyIndex.html

Subject classifications