If distances d_ij in distance matrices D express consequently with the topological distance matrices squares of Euclidean distances of corresponding vertices, then the difference scheme SDS^T, where S is the transposed incidence matrix of the graph, gives correct angles between arcs. From three nonzero eigenvalues of straight chains, the lowest one is its Wiener number.

Topological distance matrices, where the distance d_ij measures the number of arcs (edges) between vertices i and j in the graph, were used to characterize graphs in graph theory . Recently distance matrices measuring the Euclidean geometrical distances of corresponding vertices were introduced and matrices with reciprocal distance values , too.

In my previous papers , I interpreted the difference scheme SDS^T = -2I, true for trees (it was found by Rutherford ), as a symptom of orthogonality of arcs in trees. S is the transposed incidence matrix S, s _ij = -1, when the arc i goes from the vertex j, s _ij = 1, when the arc i goes to the vertex j and s _ij = 0, otherwise. I also noted that a straight chain can remain straight in spaces of all dimensions. Nevertheless, I was not able to formulate distance matrices of straight chains. Their form surprises by its simplicity.

If two consecutive orthogonal displacements lead to the distance 2, the square of the diagonal of the unit square, than two consecutive straight displacements must lead to the distance 4, square of the straight Cartesian (Euclidean) distance 2. The distances in distance matrices should be expressed as the Hilbert distances.

For straight linear chains, it gives the following matrix product. 2 is the squared length of a difference vector (.., -1, 1,...)

The other 3 distances between vertices, no arcs correspond to, we obtain similarly. On the diagonal their squared lengths appear

After normalization, we get again collinear distances. The usual topological distance matrix, where Cartesian distances seems to be, gives for missing arcs squared diagonal distances (two squares and one cube).

The difference scheme SDS^T of symmetrical square matrices must be interpreted as the quadratic form, the scalar product DS is projected onto the matrix vector S. This gives angles between the consecutive differences.

If these differences are orthogonal, then the off-diagonal elements are 0, if the off-diagonal elements are -2, then the differences are collinear. The second difference of the symmetrical square matrix is equal to its scalar product onto the difference matrix, except the negative signs and the dimensionality. The second difference of n is d²n² = 0, the second difference of n² is d²n² = 2.

If arcs do not have the unit length, then they must be normalized, e. g. for a right triangle we have consecutively

where we obtain angles 30° , 60°, and 90°, as required.

The analysis of the difference scheme shows that from 4 distances, which are involved, one is always zero (the distance of the vertex to itself). The off-diagonal elements are therefore always cosines of the corresponding angles in triangles

If we interpret distances through the arcs as the squared Euclidean geometrical distances, then we can study configurations of graphs embedded into the graph space. The trees were already mentioned, all their arcs are orthogonal. Conformations of cycles with even number of vertices are interesting. The C₄ forms a square, each arc is orthogonal with its both neighbors and collinear with the fourth arc

C₄ bent on the regular tetrahedron with the distance matrix corresponding to the distance matrix of the complete graph K₄ gives another matrix angles. Neighbor arcs form angles 60°, each arc is orthogonal with its opposite

There exist 3 embeddings of the cycle C₆ onto vertices of the cube. The first one is identical with the usual distance matrix and leads to three collinear pairs of orthogonal arcs

The two another forms have some distances shorter and lead to the another arrangement of collinear arcs.

The collinear arcs in the third conformation are (1-5), (2-4) and (3-6), respectively. The planar conformation of C₆ has the following matrix and resulting matrix of angles between bonds

where angles are 120°, 60°, 180°, 300°, and 240°, respectively. The uneven cycles have each arc orthogonal with its neighbors on both sides, but the pair of its opposites forms angles 60° to it. This conformation is obtained by a rotation of two consecutive right angles for 60 through the given arc. The result appears at arcs closing the cycle.

The distance matrices of complete graphs K_n can be expressed as D_k = JJ^T - I, where J is the unit vector column and I is the unit diagonal vector. The product

SJJ^TS^T = 0, where 0 is the zero matrix. Therefore, SD_kS = -SS^T. The outer product of the incidence matrix of a graph with simple arcs has on the diagonal 2. The off diagonal elements are either 0, the arcs do not have any common vertex, or 1, if two arcs meet in a vertex. The cosine 60° is 0.5, therefore equilateral structures appear in complete graphs. K₃ is the equilateral triangle, K₄ the equilateral tetrahedron. Six edges of the equilateral tetrahedron form three pairs of orthogonal arcs.

The quadratic form of complete graphs can be formulated in block form consecutively using the (n -1) complete graphs and unit vectors

Increasing the dimension of the complete graph, there will appear (n - 3) orthogonal arcs to each parent arc. Inserting the distance matrix of the star rooted in the nth vertex into SS^T of the complete graph, we get for the star graph the product

The arcs of the star are orthogonal. The arcs connecting its loose vertices have the double length (on the diagonal, 4 appear). These are diagonals of corresponding squares, as can be checked by calculation of cosines. 2/8^1/2 is cos 45°. The direct verification is possible only for K₄, with 3 orthogonal axes.

Distance matrices of straight chains have 3 nonzero eigenvalues: W+a, -a and -W, where W is the topological Wiener number and a has following values

The eigenvector of the lowest eigenvalue W has elements v = -1 + 2(j - 1)/(n - 1), which weight n consecutive squared numbers k from -(n-1) till (n - 1). It leads to the combinatorial identity

where x ranges from 0 till (n - 1). If chains increments are 2 vertices, then the changes between consecutive counts give a possibility to use the full induction

as [1 - 2j/(n-1)]W + (n+1-j) - j = [1 - 2j/(n-1)]W which is verified by direct calculations. For x = 0, the identity simplifies to (n - 1 - 2k)² = W. The eigenvalue a at straight chains is produced by the reflexion plane (elements of the eigenvector are symmetrical to the center of the chain) and it is a rotation tensor

The proof is simple. Sum of squared eigenvalues must be equal to the trace of the squared matrix, it means to the double sum of d

Solving the quadratic equation gives the result. Four eigenvalues (including zero) can be expressed as

We can compare 3 nonzero eigenvalues of straight linear chains with 3 distinct eigenvalues of the topological distance matrices of stars. The positive eigenvalue is the sum of all negative eigenvalues, as observed by Trinajstic et all². There are (n - 2) eigenvalues -2 and one special eigenvalue

Corresponding eigenvectors for stars rooted in v are

Due to monotony of the distance matrices, all products can be easily found. The eigenvalue a is obtained as the solution of the quadratic equation

a² + 2(n-2)a -(n-1) = 0.

The planar conformation of C₆ has following eigenvalues 12, 0, 0, 0, -6, and -6, compared with two conformations of C₆ embedded onto the cube 9, 0, 0, -1, -4, -4 and 8.4244, 0, 0, -1.4244, -3, -4 (two permutations with lesser distances).

The maximal eigenvalue of even planar cycles on the circle with unit radius is 2n and its eigenvector is the unit vector (this corresponds to n²/4 for topological distance matrices). The even distances on the circle form right triangles over the diameter as the hypotenuse and their pairs sum to 4.

It can be conjectured, that all quadratic distance matrices of graphs embedded in the three dimensional space have only 3 nonzero eigenvalues.

The given examples show a consistent picture. The elements of topological distance matrices are the Mahalanobis distances, squares of the unit Cartesian (Euclidean) distances in the multidimensional space. Squared geometrical Euclidean distances give correct angles between arcs of graphs in the 3 dimensional space, too. This consistency is a sole argument for their support against matrices with straight distances.

The question, if they will have some practical value for chemistry as a source of topological indices, remains open. The variation in linear normal alkanes with the Wiener number was already interpreted as the change of the shape of molecules from stiff rods to bent chains . In any case, geometrical distance matrices introduced by Trinajstic and by many other authors are not analogues of topological distance matrices. They have too many nonzero eigenvalues, which is an evidence of their dimensionality. Their wrong moments are, maybe, a cause why they failed in correlations with physicochemical properties.

REFERENCES AND NOTES

(1) 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.

(2) Mihalic, Z.; Veljan, D.; Amic, D.; Nicolic, S.; Plavsic,D.; Trinajstic, N. The Distance Matrix in Chemistry, J.Math. Chem., 1992, 11, 223-258.

(4) Randic, M; Jerman-Blazic, B.; Trinajstic, N. On the 3-dimensional Molecular Descriptors, Computers Chem., 1990, 14, 237-246.

(5) Bosnjak, N.; Mihalic, Z.; Trinajstic N. Application of Topographic Indices to Chromatographic Data: Calculation of Retention Indices of Alkanes, J. Chromat., 1991, 540, 30-440.

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

(7) Mihalic, Z.; Trinajstic, N. The Algebraic Modeling of Chemical Structures: On the Development of Three Dimensional Molecular Descriptors, J. Mol. Struct. (THEOCHEM)., 1991, 78, 65-78.

(8) Ivanciuc, O.; Balaban, T.S.; Balaban, A. T. Design of Topological Indexes. Part 4. Reciprocal Distance Matrix, Related Local Vertex Invariants and Topological Indexes J. Math.Chem. 1993, 12, 309-318.

(9) Kunz, M. A Mobius Inversion of the Ulam subgraph Conjecture. J. Math.Chem. 1992, 9, 297-305.

(10) Rutherford, J.S. Theoretical Prediction of Bond-Valence Networks. Acta Crystallogr. 1990, B46, 289-292.

(11) Rouvray, D.H.; Pandey, R.B. The Fractal Nature, Graph Invariants and Physicochemical Properties of Normal Alkanes J. Chem. Phys. 1986, 85, 2286-2290.