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About Eigenvalues of Quadratic Distance Matrices of Alkanes

Milan Kunz

Placed on Internet: May, 2000, improved version November 2002.

The topological distance matrices of alkanes have a key position in the development and applications of topological indices, since the sums of their elements is known as the Wiener number [1]. This index correlates well with many properties of alkanes, especially with their boiling points [2].

Trinajstic with his group tried to introduce a new index based on the geometrical distances of carbon atoms in alkanes. They calculated not only this index but as well the characteristics polynomials and eigenvalues of these quadratic matrices [3-10].

I observed that the topological distances in graphs are in fact quadratic distances in n-dimensional space and that to show the right geometry, it is necessary to use quadrates of actual distances [11,12]. Than a straight rod has only three nonzero eigenvalues.

Being lazy, I hoped that somebody will test this conclusion with alkanes. But nobody did it (at least I do not know), and I must do it myself.

I tested linear alkanes using the program MOLDA, free on Internet, to determine distances. There exists some difference between these data and data used by the Trinajstic group, the distances differ for some percent. I did not verified which distances are more correct. The used n-heptane matrix is following, lower alkanes were obtained by consecutive decreasing of matrix order:

The eigenvalues are tabulated as follows:

There are always (except propane) four nonzero eigenvalues, one positive and three negative, except values representing rounding errors. This can be explained by the fact, that the alkanes had linear form, and they are placed in a plane.

The change of conformation are changing the eigenvalues, as the example of seven conformations of hexane shows. Here I used the Trinajstic values from their paper, I only squared them.

The eigenvalues are tabulated together with the 3-dimensional Wiener number W, as calculated by the Trinajstic group:

The conformation AAA eigenvalues differ from previous slightly but again two small positive eigenvalues appear. The other conformations have five eigenvalues, since the shape of the hexanes is not planar but 3-dimensional. The seventh eigenvalues are always negligible small.

Rather interesting is the correlation between the positive eigenvalue and the 3-dimensional Wiener number. It is nearly linear, but a better correlation has the form

y = 59.5781 exp(-17.5209/x)

with r = 1 and s = 0,2023.

Similarly, different conformations of cycle C₆behave. Chemically, it can be cyclohexane as well as benzene which has planar configuration. Another planar configuration has the form of a rectangle. When we identify two vertices, we get a regular pyramide with the topological distance matrix

Its eigenvalues are tabulated together with other conformation:

Rhe eigenvalues can expressed as:

24.25 and 0.25 can be written as 12 +/- 12, 42.25 and 2.5 as 22.25 +/- 20, 38.25 and 6.25 as 22.25 +/-16.

On other place of this page is placed the paper [13], where I studied some geometrical planar objects. The results with alkanes confirm conclusions of this paper.

Literature

1. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69, (1949) 17-20.

2. Rouvray,D.H. The Topological Matrix in Quantum Chemistry.In: Balaban,A.T.(Ed.) Chemical Applications of Graph Theory, Academic Press,London,1975, 175-221.

3. B. Bogdanov, S. Nikolic and N. Trinajstic, On the three dimensional Wiener index, J. Math. Chem., 3 (1989) 299-309.

4. M. Randic, B. Jerman-Blazic and N. Trinajstic, On the 3-dimensional molecular descriptors, Computers Chem., 14 (1990) 237-246.

5. N. Bosnjak, Z. Mihalic and N. Trinajstic, Application of topographic indices to chromatographic data: calculation of retention indices of alkanes, J. Chromat., 540 (1991) 430-440.

6. Z. Mihalic, D. Veljan, D. Amic, S. Nikolic, D. Plavsic and N. Trinajstic, The distance matrix in chemistry, J. Math. Chem., 11 (1992) 223-258.

7. Z. Mihalic and N. Trinajstic, The algebraic modeling of chemical structures: on the development of three dimensional molecular descriptors, Theochem., 78 (1991) 65-78.

8. S. Nikolic, N. Trinajstic, Z. Mihalic and S. Carter, On the geometric-distance matrix and the structural invariants of molecular systems, Chem. Phys. Letters, 179 (1991) 21-28.

9. Z. Mihalic, S. Nikolic and N. Trinajstic, Comparative study of molecular descriptors derived from the distance matrix, J. Inf. Comput. Sci., 32 (1992) 28-37.

10. Z. Mihalic, D. Veljan, D. Amic, S. Nikolic, D. Plavsic and N. Trinajstic, The distance matrix in chemistry, J. Math. Chem., 11 (1992) 223-258.

11. M. Kunz: On topological and geometrical distance matrices, J. Math. Chem., 13 (1993) 145-151.

12. M. Kunz: Distance matrices yielding angles between arcs of the graphs, J. Chem. Inform. Comput. Sci., 34, (1994) 957-959.

13. M. Kunz, About eigenvalues of quadratic distance matrices of planar objects, lowest1.htm

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