Molecular Topological Index

The molecular topological index is a graph index defined by

where  are the components of the vector

with  the adjacency matrix,  the graph distance matrix, and  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).

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 , 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.

graph class	OEIS	, , ...
Andrásfai graph	A192790	4, 80, 336, 880, 1820, 3264, 5320, ...
antiprism graph	A192791	X, X, 240, 448, 760, 1200, 1792, 2560, ...
Apollonian network	A192792	72, 360, 2556, 22572, 219636, 2204244, ...
cocktail party graph	A181773	X, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, ...
complete bipartite graph	A192418	4, 48, 180, 448, 900, 1584, 2548, 3840, 5508, ...
complete graph	A181617	0, 4, 24, 72, 160, 300, 504, 784, 1152, ...
complete tripartite graph	A192491	1, 10, 36, 88, 175, 306, 490, 736, ...
crossed prism graph	A192793	X, 360, 900, 1872, 3420, 5688, 8820, ...
crown graph	A192796	X, X, 132, 360, 760, 1380, 2268, 3472, 5040, ...
cube-connected cycle graph	A192191	X, X, 5544, 57408, 458400, 3339648, 21641088, ...
cycle graph	A192797	X, X, 24, 48, 80, 132, 196, 288, ...
folded cube graph	A192826	X, 72, 448, 2400, 13824, 72128, 389120, ...
gear graph	A192827	X, X, 11, 88, 231, 440, 715, 1056, ...
grid graph	A192828	X, 48, 440, 2008, 6468, 16736, 37248, ...
grid graph	A192829	360, 8064, 68928, 355470, 1340424, 4086180, ...
halved cube graph	A192830	0, 4, 72, 672, 4800, 30240, ...
hypercube graph	A192831	4, 48, 360, 2304, 13600, 76032, 407680, ...
Möbius ladder	A192833	X, X, 180, 336, 600, 936, 1428, 2016, 2808, ...
Mycielski graph	A192834	0, 4, 80, 800, 6248, 43424, 283880, 1793600, ...
odd graph	A192835	0, 24, 540, 12040, 258300, 5258484, ...
pan graph	A192836	X, X, 14, 29, 48, 83, 126, 193, 272, 383, 510, ...
path graph	A121318	0, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, ...
permutation star graph	A192837	0, 4, 132, 4680, 214080, 12416400, ...
prism graph	A192838	X, X, 180, 360, 600, 972, 1428, 2064, 2808, ...
rook graph	A192832	X, 48, 576, 2880, 9600, 25200, 56448, 112896, ...
star graph	A016742	0, 4, 16, 36, 64, 100, 144, 196, 256, ...
sun graph	A192845	X, X, 180, 400, 740, 1224, 1876, 2720, 3780, ...
sunlet graph	A192846	X, X, 126, 256, 430, 696, 1022, 1472, ...
tetrahedral graph	A192847	7020, 30240, 100800, 281232, 687960
triangular graph	A192849	X, 0, 24, 240, 1080, 3360, 8400, 18144, ...
web graph	A192850	X, X, 414, 832, 1390, 2232, 3262, 4672,
wheel graph	A139098	X, X, X, 72, 128, 200, 288, 392, 512, ...

Closed forms are summarized in the following table.

graph
Andrásfai graph
antiprism graph
cocktail party graph
complete bipartite graph
complete graph
complete tripartite graph
crossed prism graph
crown graph
cycle graph
gear graph
grid graph
grid graph
Mycielski graph
path graph
prism graph
rook graph
star graph
sun graph
sunlet graph
triangular graph
web graph
wheel graph

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, 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."