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分子拓撲指數

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分子拓撲指數是由以下公式定義的圖指標

MTI=sum_(i=1)^nE_i,

其中 E_i 是 向量 的組成部分

E=(A+D)d,

其中 A 是鄰接矩陣，D 是圖距離矩陣，而 d 是圖的頂點度向量。分子拓撲指數僅對連通圖有明確定義；對於具有孤立節點的非連通圖是不確定的，而對於所有其他非連通圖則為無窮大。

除非另有說明，否則在計算此類指標時通常會忽略氫原子，正如有機化學家在將苯環寫成六邊形時通常所做的那樣 (Devillers and Balaban 1999, p. 25)。

MolecularTopologicalIndex440

分子拓撲指數的區分度不高，例如，上面展示的三個 10 節點非同構圖共享相同的指數值 440 (Devillers and Balaban 1999, p. 140)。事實上，四節點爪圖 和 方圖 已經無法使用該指數區分（兩者都具有指數 48），非 MTI 唯一連通圖的數量在 n=1, 2, ... 節點上由 0, 0, 0, 2, 12, 87, 815, 11086, ... (OEIS A193125) 給出。

常見圖的分子拓撲指數的預計算值在 Wolfram Language 中實現為GraphData[graph,"MolecularTopologicalIndex"].

下表總結了各種特殊圖類的分子拓撲指數值。

圖類OEISMTI(G_1), MTI(G_2), ...
Andrásfai 圖A1927904, 80, 336, 880, 1820, 3264, 5320, ...
反稜柱圖A192791X, X, 240, 448, 760, 1200, 1792, 2560, ...
阿波羅網路A19279272, 360, 2556, 22572, 219636, 2204244, ...
雞尾酒會圖 K_(n×2)A181773X, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, ...
完全二部圖 K_(n,n)A1924184, 48, 180, 448, 900, 1584, 2548, 3840, 5508, ...
完全圖 K_nA1816170, 4, 24, 72, 160, 300, 504, 784, 1152, ...
完全三部圖 K_(n,n,n)A1924911, 10, 36, 88, 175, 306, 490, 736, ...
交叉稜柱圖A192793X, 360, 900, 1872, 3420, 5688, 8820, ...
冠圖A192796X, X, 132, 360, 760, 1380, 2268, 3472, 5040, ...
立方體連線環圖A192191X, X, 5544, 57408, 458400, 3339648, 21641088, ...
環圖 C_nA192797X, X, 24, 48, 80, 132, 196, 288, ...
摺疊立方體圖A192826X, 72, 448, 2400, 13824, 72128, 389120, ...
齒輪圖A192827X, X, 11, 88, 231, 440, 715, 1056, ...
網格圖 P_n square P_nA192828X, 48, 440, 2008, 6468, 16736, 37248, ...
網格圖 P_n square P_n square P_nA192829360, 8064, 68928, 355470, 1340424, 4086180, ...
半立方體圖A1928300, 4, 72, 672, 4800, 30240, ...
超立方體圖 Q_nA1928314, 48, 360, 2304, 13600, 76032, 407680, ...
莫比烏斯梯子圖 M_nA192833X, X, 180, 336, 600, 936, 1428, 2016, 2808, ...
Mycielski 圖A1928340, 4, 80, 800, 6248, 43424, 283880, 1793600, ...
奇圖 O_nA1928350, 24, 540, 12040, 258300, 5258484, ...
平底鍋圖A192836X, X, 14, 29, 48, 83, 126, 193, 272, 383, 510, ...
路徑圖 P_nA1213180, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, ...
置換星圖 PS_nA1928370, 4, 132, 4680, 214080, 12416400, ...
稜柱圖 Y_nA192838X, X, 180, 360, 600, 972, 1428, 2064, 2808, ...
車圖 K_n square K_nA192832X, 48, 576, 2880, 9600, 25200, 56448, 112896, ...
星圖 S_nA0167420, 4, 16, 36, 64, 100, 144, 196, 256, ...
太陽圖A192845X, X, 180, 400, 740, 1224, 1876, 2720, 3780, ...
日瓣圖 C_n circledot K_1A192846X, X, 126, 256, 430, 696, 1022, 1472, ...
四面體約翰遜圖A1928477020, 30240, 100800, 281232, 687960
三角形圖A192849X, 0, 24, 240, 1080, 3360, 8400, 18144, ...
網狀圖A192850X, X, 414, 832, 1390, 2232, 3262, 4672,
輪圖 W_nA139098X, X, X, 72, 128, 200, 288, 392, 512, ...

閉合形式總結在下表中。

圖MTI(G_n)
Andrásfai 圖2n(3n-2)(3n-1)
反稜柱圖4n(n^2+n+8)
雞尾酒會圖 K_(n×2)8(n-1)n(2n-1)
完全二部圖 K_(n,n)4n^2(2n-1)
完全圖 K_n2n(n-1)^2
完全三部圖 K_(n,n,n)12n^2(3n-1)
交叉稜柱圖12n(2n^2+7)
冠圖2(n-1)n(4n-1)
環圖 C_n1/4n[2n^2+(-1)^n+15]
齒輪圖11n(3n-2)
網格圖 P_n square P_n2/3(n+2)(4n^4-13n^3+25n^2-24n+6)
網格圖 P_n square P_n square P_nn(6n^6-7n^5-3n^4+4n^3+36n^2-66n+24)
Mycielski 圖-1/(36)(3·2^n-8)(27·2^n-143^n-18)
路徑圖 P_n1/3(2n^3+3n^2+13n-6)
稜柱圖 Y_n3/2n(11+(-1)^n+2n(2+n))
車圖4(-1+n)^2n^2(1+n)
星圖 S_n4(n-1)^2
太陽圖4n(n^2+3n-3)
日瓣圖 C_n circledot K_1n[2n(n+3)+(-1)^n+7]
三角形圖(n-2)^2n(n^2-1)
網狀圖n[6n^2+22n+3((-1)^n+7)]
輪圖 W_n8(n-1)^2

另請參閱

鄰接矩陣, 度序列, 圖距離矩陣, 拓撲指數, 頂點度

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參考文獻

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 中被引用

分子拓撲指數

請引用為

Weisstein, Eric W. "Molecular Topological Index." From --A Resource. https://mathworld.tw/MolecularTopologicalIndex.html

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