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 Johnson 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.