Graph-theoretical method for evaluation of eigenvalues of real symmetric matrices from an eigenvector approach

TLDR: In this paper, it was shown that for any two topologically similar vertices i and k, A2ji=A2jk; for all j≠k and i where Aji is the cofactor of the (j,i) element of the secular determinant det(xI-A), A is the adjacency matrix of the graph G and I is the (n×n) unit matrix.

Abstract: It has been proved that if {vi}, i= 1–n, is the set of vertices of an undirected labelled graph G, then for any two topologically similar vertices i and k, A2ji=A2jk; for all j≠k and i where Aji is the cofactor of the (j,i) element of the secular determinant det(xI–A), A is the adjacency matrix of the graph G and I is the (n×n) unit matrix.As any real symmetric matrix, A, can be represented by an undirected vertex- and edge-weighted graph (G), the above relation has been utilised, in conjunction with a recently developed graph-theoretical method for expressing eigenvectors of A as polynomials in terms of eigenvalues, to determine a good number of eigenvalues of the matrix. The method, for the first time, utilises a newly developed technique of determination of eigenvectors for evaluation of eigenvalues. In one particular case it has been shown that the present method can reduce the required polynomial equations to a degree lower than that possible by McClelland's technique for factorisation of chemical graphs. Some applications of the method (other than HMO theory), for example, calculation of principal stress tensors in fluid dynamics and force constants in a molecular vibration problem, are illustrated.