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Molecular Topological Index

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The molecular topological index is a graph index defined by

MTI=sum_(i=1)^nE_i,

where E_i are the components of the vector

E=(A+D)d,

with A the adjacency matrix, D the graph distance matrix, and d the vector of vertex degrees of a graph. The molecular topological index is well-defined only for connected graphs, being indeterminate for disconnected graphs having isolated nodes and infinity for all other disconnected graphs.

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).

MolecularTopologicalIndex440

The molecular topological index is not very discriminant, with the three 10-node nonisomorphic graphs illustrated above for example sharing the same index value of 440 (Devillers and Balaban 1999, p. 140). In fact, the paw graph and square graph on four nodes are already indistinguishable using the index (both have index 48), with the number of non-MTI-unique connected graphs on n=1, 2, ... nodes given by 0, 0, 0, 2, 12, 87, 815, 11086, ... (OEIS A193125).

Precomputed values of the molecular topological index for common graphs are implemented in the Wolfram Language as GraphData[graph, "MolecularTopologicalIndex"].

The following table summarizes values of the molecular topological index for various special classes of graphs. The index of each value list in the table starts at 1. An X indicates that the family or its molecular topological index is not defined for the corresponding parameter value. Some cited OEIS sequences extend a formula to parameters outside the family's domain, while this table lists the values for the graphs themselves.

graph classOEISMTI(G_1), MTI(G_2), ...
Andrásfai graphA1927904, 80, 336, 880, 1820, 3264, 5320, ...
antiprism graphA192791X, X, 240, 448, 760, 1200, 1792, 2560, ...
Apollonian networkA19279272, 360, 2556, 22572, 219636, 2204244, ...
cocktail party graph K_(n×2)A181773X, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, ...
complete bipartite graph K_(n,n)A1924184, 48, 180, 448, 900, 1584, 2548, 3840, 5508, ...
complete graph K_nA1816170, 4, 24, 72, 160, 300, 504, 784, 1152, ...
complete tripartite graph K_(n,n,n)A19249124, 240, 864, 2112, 4200, 7344, 11760, 17664, ...
crossed prism graphA192793X, 360, 900, 1872, 3420, 5688, 8820, ...
crown graphA192796X, X, 132, 360, 760, 1380, 2268, 3472, 5040, ...
cube-connected cycle graphA192191X, X, 5544, 57408, 458400, 3339648, 21641088, ...
cycle graph C_nA192797X, X, 24, 48, 80, 132, 196, 288, ...
folded cube graphA192826X, 72, 448, 2400, 13824, 72128, 389120, ...
gear graphA192827X, X, 231, 440, 715, 1056, 1463, 1936, ...
grid graph P_n square P_nA1928280, 48, 440, 2008, 6468, 16736, 37248, ...
grid graph P_n square P_n square P_nA1928290, 360, 8064, 68928, 355470, 1340424, 4086180, ...
halved cube graphA1928300, 4, 72, 672, 4800, 30240, ...
hypercube graph Q_nA1928314, 48, 360, 2304, 13600, 76032, 407680, ...
Möbius ladder M_nA192833X, X, 180, 336, 600, 936, 1428, 2016, 2808, ...
Mycielski graphA1928340, 4, 80, 800, 6248, 43424, 283880, 1793600, ...
odd graph O_nA1928350, 24, 540, 12040, 258300, 5258484, ...
pan graphA192836X, X, 48, 83, 126, 193, 272, 383, 510, 677, 864, ...
path graph P_nA1213180, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, ...
permutation star graph PS_nA1928370, 4, 132, 4680, 214080, 12416400, ...
prism graph Y_nA192838X, X, 180, 360, 600, 972, 1428, 2064, 2808, ...
rook graph K_n square K_nA1928320, 48, 576, 2880, 9600, 25200, 56448, 112896, ...
star graph S_nA0167420, 4, 16, 36, 64, 100, 144, 196, 256, ...
sun graphA192845X, X, 180, 400, 740, 1224, 1876, 2720, 3780, ...
sunlet graph C_n circledot K_1A192846X, X, 126, 256, 430, 696, 1022, 1472, ...
tetrahedral Johnson graphA1928477020, 30240, 100800, 281232, 687960, ...
triangular graphA192849X, 0, 24, 240, 1080, 3360, 8400, 18144, ...
web graphA192850X, X, 414, 832, 1390, 2232, 3262, 4672, ...
wheel graph W_nA139098X, X, X, 72, 128, 200, 288, 392, 512, ...

Closed forms are summarized in the following table. The formulas for path graphs and grid graphs apply for n>=2. Their n=1 members are singleton graphs and have index 0.

graphMTI(G_n)
Andrásfai graph2n(3n-2)(3n-1)
antiprism graph4n(n^2+n+8)
cocktail party graph K_(n×2)8(n-1)n(2n-1)
complete bipartite graph K_(n,n)4n^2(2n-1)
complete graph K_n2n(n-1)^2
complete tripartite graph K_(n,n,n)12n^2(3n-1)
crossed prism graph12n(2n^2+7)
crown graph2(n-1)n(4n-1)
cycle graph C_n1/4n[2n^2+(-1)^n+15]
gear graph11n(3n-2)
grid graph P_n square P_n2/3(n+2)(4n^4-13n^3+25n^2-24n+6)
grid graph P_n square P_n square P_nn(6n^6-7n^5-3n^4+4n^3+36n^2-66n+24)
Mycielski graph-1/(36)(3·2^n-8)(27·2^n-143^n-18)
path graph P_n1/3(2n^3-3n^2+13n-18)
prism graph Y_n3/2n(11+(-1)^n+2n(2+n))
rook graph4(-1+n)^2n^2(1+n)
square graph Sq_n4(n-1)^2n^2(n+1)
star graph S_n4(n-1)^2
sun graph4n(n^2+3n-3)
sunlet graph C_n circledot K_1n[2n(n+3)+(-1)^n+7]
triangular graph(n-2)^2n(n^2-1)
web graphn[6n^2+22n+3((-1)^n+7)]
wheel graph W_n8(n-1)^2

See also

Adjacency Matrix, Degree Sequence, Graph Distance Matrix, Molecular Graph, Topological Index, Vertex Degree

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References

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, A121318, A139098, 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."

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Molecular Topological Index

Cite this as:

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

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