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哈拉里指数

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图 G 在 n 个顶点上的哈拉里指数由 Plavšić 等人 (1993) 定义为

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

其中

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

是 图距离矩阵 D 的倒数 (Plavšić 等人 1993; Devillers 和 Balaban, p. 80, 2000)。

需要注意，因为有些作者包含了前导因子 1/2 (例如，Plavšić 等人 1993, Mercader 等人 2001)，而另一些作者则省略了它 (例如，Devillers 和 Balaban 1999, pp. 111 和 202)。

除非另有说明，否则在计算此类指标时通常会忽略氢原子，正如有机化学家在将苯环写成六边形时通常所做的那样 (Devillers 和 Balaban 1999, p. 25)。

下表总结了各种特殊图类的哈拉里指数值。

图类OEISH(G_1), H(G_2), ...
Andrásfai 图A000000/A0000001, 15/2, 20, 77/2, 63, 187/2, 130, 345/2, 221, ...
反棱柱图A000000/A000000X, X, 27/2, 22, 95/3, 42, 637/12, 194/3, 384/5, ...
阿波罗尼安网络A000000/A0000006, 18, 80, 470, 3248, 122106/5, 3394391/20, 6406407/20, ...
bishop 图 B_(n,n)A2961970, 2, 13, 42, 102, 208, 379, 636, 1004, 1510, ...
black bishop 图 BB_(n,n)A2961980, 1, 8, 21, 55, 104, 197, 318, 514, 755, ...
鸡尾酒会图 K_(n×2)A000000/A0000000, 5, 27/2, 26, 85, 126, 175, 232, 297, 370, ...
完全二部图 K_(n,n)A0003262, 5, 12, 44, 70, 102, 140, 184, ...
完全图 K_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ...
完全三部图 K_(n,n,n)A000000/A0000003, 27/2, 63, 114, 180, 261, ...
2n-交叉棱柱图A000000/A00000058/3, 39, 368/3, 514/3, 1116/5, 4166/15, 35128/105, ...
冠状图A000000/A000000X, X, 10, 58/3, 95/3, 47, 196/3, 260/3, 111, 415/3, ...
超立方体连接环图A000000/A000000X, X, 556/5, 57376/105, 162634/63, 34149904/3003, ...
环图 C_nA160046/A160047X, X, 3, 5, 15/2, 10, 77/6, 47/3, 75/4, 131/6, ...
斐波那契立方体图A000000/A0000001, 5/2, 22/3, 71/4, 216/5, 1219/12, 25033/105, ...
折叠立方体图A000000/A000000X, 1, 6, 22, 80, 808/3, 2800/3, 9488/3, 11072, ...
齿轮图A000000/A000000X, X, 29/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ...
网格图 P_n square P_nA296191/A2961920, 5, 133/6, 293/5, 3399/28, 137111/630, 140351/396, ...
网格图 P_n square P_n square P_nA000000/A0000000, 58/3, 2402/15, 30617/45, 7168769/3465, ...
半立方体图A290347/A2903480, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, ...
河内图A000000/A0000003, 22, 4276/35, 1835837/3003, 175359949924361/60168147039, ...
舵图A000000/A00000029/2, 133/6, 125/4, 167/4, 161/3, 67, 327/4, ...
超立方体图 Q_nA290343/A2903441, 5, 58/3, 206/3, 3548/15, 12136/15, 291824/105, ...
Keller 图 K_nA2961890, 80, 1552, 27264, 460544, 7634944, ...
王图 Ki_(n,n)A1449450, 6, 28, 76, 160, 290, 476, 728, 1056, 1470, ...
马图 Kn_(n,n)A000000/A0000000, 0, 47/3, 309/5, 150, 1769/6, 7724/15, 24733/30, ...
Menger 海绵图A000000/A0000001147/15, 207460203161/19684665, ...
莫比乌斯梯子图A000000/A000000X, X, 12, 20, 85/3, 38, 287/6, 176/3, 348/5, 244/3, ...
Mycielski 图A296193/A0000000, 1, 15/2, 75/2, 162, 1317/2, 2610, 20505/2, 40212, ...
奇图 O_nA0000000, 3, 30, 280, 2730, 57057/2, 635635/2, ...
平底锅图A000000/A000000X, X, 5, 22/3, 61/6, 155/12, 16, 571/30, 1339/60, ...
路径图 P_nA160048/A1600490, 2, 5, 26/3, 77/6, 87/5, 223/10, 962/35, ...
排列星图 PS_nA296190/A2960570, 1, 10, 123, 2202, 59040, 2287680, 121394000, ...
棱柱图 Y_nA000000/A000000X, X, 12, 58/3, 85/3, 75/2, 287/6, 874/15, ...
后图 Q_(n,n)A2961960, 6, 32, 98, 230, 460, 826, 1372, 2148, 3210, ...
车补图 K_n square K_n^_A0923640, 2, 27, 96, 250, 540, 1029, 1792, 2916, 4500, ...
车图 K_n square K_nA085740X, 5, 54, 168, 400, 810, 1470, 2464, 3888, 5850, ...
Sierpiński 地毯图A000000/A00000047/3, 23255059/51480, ...
Sierpiński 垫片图A000000/A0000003, 12, 227/4, 5553/20, 161390213/120120, ...
Sierpiński 四面体图A000000/A0000006, 69/2, 1055/4, 599803/280, 279423163/16016, ...
星图 S_nA160050/A1306580, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ...
太阳图A000000/A000000X, X, 10, 97/6, 95/4, 158/5, 2429/60, 743/15, ...
日瓣图 C_n circledot K_1A000000/A000000X, X, 10, 97/3, 95/2, 316/5, 2429/30, 1486/15, 594/5, ...
四面体 Johnson 图A000000/A000000X, X, 415/3, 2345/6, 2800/3, 1981, 3850, 6985, 11990, ...
环面网格图 C_n square C_nA000000/A000000X, X, 27, 206/3, 875/6, 1287/5, 12691/30, 66964/105, ...
转置图A2961940, 1, 12, 162, 3010, 81000, 3105396, 162469104, ...
三角图A000000/A000000X, 0, 3, 27/2, 75/2, 165/2, 315/2, 273, 441, 675, 990, ...
三角网格图A0274803, 12, 30, 60, 105, 168, 252, 360, 495, 660, ...
网状图A000000/A000000X, X, 45/2, 217/6, 635/12, 703/10, 1799/20, 110, ...
轮图 W_nA000000/A0000006, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51, 117/2, ...
white bishop 图 WB_(n,n)A2962001, 5, 21, 47, 104, 182, 318, 490, 755, ...

下表总结了一些特殊图类的闭合形式。 这里，H_n 是 调和数，C_n 是 卡塔兰数，Phi(z,s,a) 是 Lerch 超越函数，_pF_q 是 广义超几何函数，并且 s(n,m) 是 第一类斯特林数。

图哈拉里指数
Andrásfai 图1/2(3n-1)(2n-1)
反棱柱图{2(2nH_(n/2-1)+3) for n even; 2n(2H_((n-1)/2)+1/(n+1)) for n odd
bishop 图 B_(n,n)1/(48)[6n^4+16n^3-36n^2+8n+3-3(-1)^n]
black bishop 图 BB_(n,n)1/(96)[6n^4+16n^3-30n^2+20n-15-3(-1)^n(2n^2+4n-5)]
鸡尾酒会图 K_(n×2)1/2n(4n-3)
完全二部图 K_(m,n)1/4(m^2+n^2-m-n)+mn
完全二部图 K_(n,n)1/2n(3n-1)
完全图 K_n1/2n(n-1)
完全三部图 K_(n,n,n)3/4n(5n-1)
2n-交叉棱柱图(8H_(n-1)-1/3)n+4
冠状图 K_2 square K_n^_1/6n(9n-7)
环图 C_n1/2(1+(-1)^n)+nH_(|_(n-1)/2_|)
空图 K^__n0
齿轮图 G_n1/(24)n(17n+65)
网格图 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]
半立方体图 1/2Q_n-2^(n-1)H_n-2^(2n-1)R[Phi(2,1,n+1)]
舵图1/(24)n(17n+65)
超立方体图 Q_n2^(n-1)n_3F_2(1,1,1-n;2,1;-1)
Keller 图{0 for n=1; 4^n[4^n-1/2(3^n+n+1)]/2 otherwise
王图 Ki_(n,n)1/3(n-1)n(5n-1)
莫比乌斯梯子图 M_n{n(4H_(n/2)-1) for n even; n(4H_((n-1)/2)-(n-3)/(n+1)) for n odd
Mycielski 图 M_n{0 for n=1; 1/(48)(36-452^n+283^(n-1)+274^(n-1)) otherwise
平底锅图{(n+2)H_(n/2)+2/(n+1) for n even; (n+2)H_((n-1)/2)+4/(n+1)-1 for n odd
路径图 P_nn(H_n-1)
棱柱图 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
后图 Q_(n,n)1/(12)(n-1)n(3n^2+13n-2)
车补图 K_n square K_n^_1/2(n-1)n^3
车图 K_n square K_n1/4(n-1)n^2(n+3)
星图 S_n1/4(n+2)(n-1)
太阳图1/6n(7n+3)
日瓣图 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
四面体 Johnson 图1/(432)(n-3)(n-2)(n-1)n(2n^2+9n+40)
环面网格图 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
转置图1/2sum_(k=1)^(n-1)((-1)^ks(n,n-k))/k
三角图1/(16)(n-2)(n-1)n(n+5)
三角网格图1/2n(n+1)(n+2)
网状图{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
轮图 W_n1/4(n-1)(n+4)
white bishop 图 WB_(n,n)1/(96)[6n^4+16n^3-42n^2-4n+21+3(-1)^n(2n^2+4n-7)]

另请参阅

图距离矩阵, 拓扑指数

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参考资料

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."

在 Wolfram|Alpha 中被引用

哈拉里指数

引用为

Weisstein, Eric W. "哈拉里指数。" 来自 MathWorld-- Wolfram Web 资源。 https://mathworld.net.cn/HararyIndex.html

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