Matrix Interpretation of the Randic Number

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Matrix Interpretation of the Randic Number

Milan Kunz

Abstract

The Randic number is one half of the sum of the off-diagonal elements of the normalized Laplace-Kirchhoff matrix. Some aspects of this fact are disscussed.

Introduction

The Randic number [1] is one from the most usefull topological indices. It was generalized by Kier as the index c with the power index ^-1/2. There remains open the formal relation of the Randic number to the other topological indices, for example to the Wiener number W.

The Laplace-Kirchhoff matrix is the quadratic form S^TS of the incidence matrix S (s_ij = -1, if the arc i goes from the vertex j, s_ij = 1, if the arc i goes ends in the vertex j, s_ij = 0, otherwise).

The quadratic form S^TS is split into the diagonal matrix of the vertex degrees V, and the off-diagonal adjacency matrix A:

S^TS = V - A.

Even before topological indices, the eigenvalues of the adjacency matrices A of aromatic compounds were connected directly with their physical properties by the Hueckel theory of molecular orbitals.

Indirectly, the adjacency matrices A of aliphatic compounds are connected with their bulk physical properties, too, since the sum of their inversed eigenvalues [2] gives the Wiener number W.

The Wiener number W is usually connected one half of the sum of the off-diagonal elements of the distance matrix D but it appears as the trace [2] of the other quadratic form SST of the incidence matrix S, and the doubled Wiener number 2W as the trace [4] of the generalized inverse of the normalized Laplace-Kirchhoff matrix.

As an analogy with the distance matrix D, the Randic number is obtained formally as one half of the sum of the off-diagonal elements of the normalized Laplace-Kirchhoff matrix

V^-1/2S^TSV^-1/2 = V^-1/2VV^-1/2 - V^-1/2AV^-1/2 = I - V^-1/2AV^-1/2.

For example, 2-methylbutane:

STS

2	-1	0	0	-1
-1	3	-1	-1	0
0	-1	1	0	0
0	-1	0	1	0
-1	0	0	0	1

VS^TSV^-1/2

1	-6^-1/2	0	0	-2^-1/2
6^-1/2	1	3^-1/2	3^-1/2	0
0	3^-1/2	1	0	0
0	3^-1/2	0	1	0
-2^-1/2	0	0	0	1

The Randic number is n at the regular graphs (even disconnected [2]), and it has minimum at stars.

At acyclic graphs, it is possible to find the other normalized quadratic form: SVS^T, since the inverse of SS^T has the form W^TW, where W is the walk (path) matrix*.

At the regular graphs, there exist the generalized inverse of the normalized Laplace-Kirchhoff matrix

V^-1/2S^TSV^-1/2

having the form

V^1/2 (S d _j(S^TS)^-1V^1/2

where d _j(S^TS)^-1 is the inverse of the matrix of the Ulam subgraph (j-th row and column deleted) of the parent graph since

V^-1/2S^TSV^-1/2V^1/2**V^1/2 = V^-1/2S^TS²**V^1/2 = V^-1/2(nI - JJ^T)V^1/2.

If all values v_ij are equal, multiplications with V^-1/2 from the left cancel multiplications with V^1/2 from the right

AV^-1/2 = I - V^-1/2AV^-1/2.

The relation between the Kier powers and traces of the corresponding matrices is in the following table:

Power of c	-1	-1/2	0	1/2	1
Elements of V^-1/2S^TSV^-1/2	1/v	1	v	v²	v³
Trace of V^-1/2S^TSV^-1/2	S 1/v	n	2m	S v²	S v³

m is the number of arcs.

There appear new problems, for example to study eigenvalues of the normalized Laplace-Kirchhoff matrices. Similarly, the Balaban index J is constructed as the product of the adjacency matrix with diagonal matrices Q^-1/2 from both sides, where q_j are distances of the vertex j to all other vertices. But these distances are derived from the other quadratic form of walk matrices WW^T [4] and their sum is sum of inversed eigenvalues of the Laplace-Kirchhoff matrix. Nobody studied what such multiplication makes with eigenvalues.

Many authors tried orthogonality of different topological indices [10]. But either two topological indices characterize a whole molecule and then they are both differently rotated forms of one matrix vector or they express only some of its properties and then they must be complementary, at least from a part. If a matrix vector is inverse to the other then they can not be orthogonal unless one has at least one zero eigenvalue and the other is infinite. But such solutions are unacceptable and we must replace orthogonality by another properties as in the case of bond and edge erasures by the complementarity inside of complete bipartite graphs.

There are two ways how to approach our problems. One possibility is based on abstract ideas of metric spaces and exploits sophisticated techniques of formal mathematics [11].The other is founded on real objects as molecules are and graphs seems to be. It tries to investigate all hidden niches of the vector space and their secrets. Its methods are cumbersome but they give insight how the space we live in is built. There appear several surprises because properties of high dimensional spaces are contraintuitive [11].

The incidence matrices S and G have just two unit elements in each row. Despite it their wreath product symmetry is complicated enough to keep us busy for long time before we learn to spell all God's names.

References

[1] M. Randic, J Math. Chem., 7{1991}155.

[2] M. Kunz, Collect. Czech. Chem. Commun., 54 {1989}2148.

[3] H. Wiener, J. Amer. Chem. Soc., 69 {1947}17.

[4] M. Kunz, J. Math. Chem., 13 {1993}145.

[5] B. Mohar in Stud. Phys. Theor. Chem., 63 MATH/CHEM/COMP 1988 ed A. Graovac/Elsevier,,Amsterdam, 1989/1.

[6] W. Heissenberg in The Physicist's Conception of Nature ed. J. Mehra /D.Reidel, Dortrecht, 1968/267.

[7] J. Dugundji, I. Ugi, Topics Curr. Chem. 39 {1973}19.

[8] D. H. Rouvray in Chemical Applications of Graph Theory ed A. T. Balaban /Academic Press, London,1976/175.

[9] M. Kunz, Coll. Czech. Chem. Commun. 55 {1990}630.

[10] K. Kovacevic, D. Plavsic, N. Trinajstic, D. Horvat in Studies in Phys. Theor. Chem., 63 MATH/CHEM/COMP 1988, ed. A. Graovac /Elsevier,Amsterdam,1989/213.

[11] P. G. Mezey in Studies in Phys. Theor. Chem., 51, Graph Theory and Topology ed. R. B. King, D. H. Rouvray /Elsevier,Amsterdam, 1987/91.