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分子拓扑指数

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分子拓扑指数是由以下公式定义的图指标

MTI=sum_(i=1)^nE_i,

其中 E_i 是 向量 的组成部分

E=(A+D)d,

其中 A 是邻接矩阵，D 是图距离矩阵，而 d 是图的顶点度向量。分子拓扑指数仅对连通图有明确定义；对于具有孤立节点的非连通图是不确定的，而对于所有其他非连通图则为无穷大。

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 and Balaban 1999, p. 25)。

MolecularTopologicalIndex440

分子拓扑指数的区分度不高，例如，上面展示的三个 10 节点非同构图共享相同的指数值 440 (Devillers and Balaban 1999, p. 140)。事实上，四节点爪图 和 方图 已经无法使用该指数区分（两者都具有指数 48），非 MTI 唯一连通图的数量在 n=1, 2, ... 节点上由 0, 0, 0, 2, 12, 87, 815, 11086, ... (OEIS A193125) 给出。

常见图的分子拓扑指数的预计算值在 Wolfram Language 中实现为GraphData[graph,"MolecularTopologicalIndex"].

下表总结了各种特殊图类的分子拓扑指数值。

图类OEISMTI(G_1), MTI(G_2), ...
Andrásfai 图A1927904, 80, 336, 880, 1820, 3264, 5320, ...
反棱柱图A192791X, X, 240, 448, 760, 1200, 1792, 2560, ...
阿波罗网络A19279272, 360, 2556, 22572, 219636, 2204244, ...
鸡尾酒会图 K_(n×2)A181773X, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, ...
完全二部图 K_(n,n)A1924184, 48, 180, 448, 900, 1584, 2548, 3840, 5508, ...
完全图 K_nA1816170, 4, 24, 72, 160, 300, 504, 784, 1152, ...
完全三部图 K_(n,n,n)A1924911, 10, 36, 88, 175, 306, 490, 736, ...
交叉棱柱图A192793X, 360, 900, 1872, 3420, 5688, 8820, ...
冠图A192796X, X, 132, 360, 760, 1380, 2268, 3472, 5040, ...
立方体连接环图A192191X, X, 5544, 57408, 458400, 3339648, 21641088, ...
环图 C_nA192797X, X, 24, 48, 80, 132, 196, 288, ...
折叠立方体图A192826X, 72, 448, 2400, 13824, 72128, 389120, ...
齿轮图A192827X, X, 11, 88, 231, 440, 715, 1056, ...
网格图 P_n square P_nA192828X, 48, 440, 2008, 6468, 16736, 37248, ...
网格图 P_n square P_n square P_nA192829360, 8064, 68928, 355470, 1340424, 4086180, ...
半立方体图A1928300, 4, 72, 672, 4800, 30240, ...
超立方体图 Q_nA1928314, 48, 360, 2304, 13600, 76032, 407680, ...
莫比乌斯梯子图 M_nA192833X, X, 180, 336, 600, 936, 1428, 2016, 2808, ...
Mycielski 图A1928340, 4, 80, 800, 6248, 43424, 283880, 1793600, ...
奇图 O_nA1928350, 24, 540, 12040, 258300, 5258484, ...
平底锅图A192836X, X, 14, 29, 48, 83, 126, 193, 272, 383, 510, ...
路径图 P_nA1213180, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, ...
置换星图 PS_nA1928370, 4, 132, 4680, 214080, 12416400, ...
棱柱图 Y_nA192838X, X, 180, 360, 600, 972, 1428, 2064, 2808, ...
车图 K_n square K_nA192832X, 48, 576, 2880, 9600, 25200, 56448, 112896, ...
星图 S_nA0167420, 4, 16, 36, 64, 100, 144, 196, 256, ...
太阳图A192845X, X, 180, 400, 740, 1224, 1876, 2720, 3780, ...
日瓣图 C_n circledot K_1A192846X, X, 126, 256, 430, 696, 1022, 1472, ...
四面体约翰逊图A1928477020, 30240, 100800, 281232, 687960
三角形图A192849X, 0, 24, 240, 1080, 3360, 8400, 18144, ...
网状图A192850X, X, 414, 832, 1390, 2232, 3262, 4672,
轮图 W_nA139098X, X, X, 72, 128, 200, 288, 392, 512, ...

闭合形式总结在下表中。

图MTI(G_n)
Andrásfai 图2n(3n-2)(3n-1)
反棱柱图4n(n^2+n+8)
鸡尾酒会图 K_(n×2)8(n-1)n(2n-1)
完全二部图 K_(n,n)4n^2(2n-1)
完全图 K_n2n(n-1)^2
完全三部图 K_(n,n,n)12n^2(3n-1)
交叉棱柱图12n(2n^2+7)
冠图2(n-1)n(4n-1)
环图 C_n1/4n[2n^2+(-1)^n+15]
齿轮图11n(3n-2)
网格图 P_n square P_n2/3(n+2)(4n^4-13n^3+25n^2-24n+6)
网格图 P_n square P_n square P_nn(6n^6-7n^5-3n^4+4n^3+36n^2-66n+24)
Mycielski 图-1/(36)(3·2^n-8)(27·2^n-143^n-18)
路径图 P_n1/3(2n^3+3n^2+13n-6)
棱柱图 Y_n3/2n(11+(-1)^n+2n(2+n))
车图4(-1+n)^2n^2(1+n)
星图 S_n4(n-1)^2
太阳图4n(n^2+3n-3)
日瓣图 C_n circledot K_1n[2n(n+3)+(-1)^n+7]
三角形图(n-2)^2n(n^2-1)
网状图n[6n^2+22n+3((-1)^n+7)]
轮图 W_n8(n-1)^2

另请参阅

邻接矩阵, 度序列, 图距离矩阵, 拓扑指数, 顶点度

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参考文献

Balaban, A. T.; Motoc, I.; Bonchev, D.; and Mekenyan, O. "Topological Indices for Structure-Activity Correlations." Top. Curr. Chem. 114, 21-55, 1983.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 30-31, 138-141, and 210-212, 1999.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.Mueller, W. R.; Szymanski, K.; Knop, J. V.; and Trinajstić, N. "Molecular Topological Index." J. Chem. Inf. Comput. Sci. 30, 160-163, 1990.Randić, M. "In Search of Structural Invariants." J. Math. Chem. 9, 97-146, 1992.Schultz, H. P. "Topological Organic Chemistry. 1. Graph Theory and Topological Indices of Alkanes." J. Chem. Inf. Comput. Sci. 29, 227-228, 1989.Schultz, H. P.; Schultz, E. B.; and Schultz, T. P. "Topological Organic Chemistry. Part 2. Graph Theory, Matrix Determinants and Eigenvalues, and Topological Indices of Alkanes." J. Chem. Inf. Comput. Sci. 30, 27-29, 1990.Sloane, N. J. A. Sequences A016742, A139098, A121318, A181617, A181773, A192191, A192418, A192491, A192790, A192791, A192792, A192793, A192796, A192797, A192826, A192827, A192828, A192829, A192830, A192831, A192832, A192833, A192834, A192835, A192836, A192837, A192838, A192839, A192845, A192846, A192847, A192848, A192849, A192850, and A193125 in "The On-Line Encyclopedia of Integer Sequences."

在 Wolfram|Alpha 中被引用

分子拓扑指数

请引用为

Weisstein, Eric W. "Molecular Topological Index." From MathWorld--A Wolfram Web Resource. https://mathworld.net.cn/MolecularTopologicalIndex.html

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