Two new graph-theoretical methods, (A) and (B), have been devised for generation of eigenvectors of weighted and unweighted chemical graphs. Both the methods show that not only eigenvalues but also eigenvectors have full combinatorial (graph-theoretical) content. Method (A) expresses eigenvector components in terms of Ulam’s subgraphs of the graph. For degenerate eigenvalues this method fails, but still the expressions developed yield a method for predicting the multiplicities of degenerate eigenvalues in the graph-spectrum. Some well-known results about complete graphs (K_n) and annulenes (C_n), viz. (i)K_n has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) C_n cannot show more than two-fold degeneracy, can be proved very easily by employing the eigenvector expression developed in method (A). Method (B) expresses the eigenvectors as analytic functions of the eigenvalues using the cofactor approach. This method also fails in the case of degenerate eigenvalues but can be utilised successfully in case of accidental degeneracies by using symmetry-adapted linear combinations. Method (B) has been applied to analyse the trend in charge-transfer absorption maxima of the some molecular complexes and the hyperconjugative HMO parameters of the methyl group have been obtained from this trend.

The change in charge-transfer absorption maxima of the π-type molecular complexes formed between chloranil and a series of phenols with varying number and position of the phenolic OH group(s) is shown to follow a certain trend. For this purpose the Coulson-Longuet-Higgins method in HMO formalism is used. The necessary eigenvector components are computed by employing a recently developed graph-theoretical technique, which avoids the calculation of the unnecessary eigenvectors. From the trends in the charge-transfer bands, the perturbational HMO parametersh andk for the phenolic oxygen and the C(sp²)-0 bond, respectively, were obtained in a straightforward way and found to be in good agreement with previously recommended values, determined by other methods.

A graph-theoretical formula for computation of products of eigen coefficientsC_rjC_sj(r≠s) of a graph has been derived. A general formula for the characteristic polynomial of a linear chain having a branch with a weighted pendant vertex has been given. These have been utilised, together with the Ulam subgraph method for obtaining squares of eigen coefficients, to take into account the perturbations in Coulomb and resonance integrals due to introduction of heteroatoms in a conjugated system or due to attachment of Me groups to conjugated atoms. Heteroatom Hückel parametersh andk of N and Me groups have been estimated and some structural information about a number of electron donors have been extracted by applying the results to the trends in the charge-transfer bands of their π-type molecular complexes.

El-Basil's Fibonacci triad (FT) method has been utilised to construct the characteristic polynomials of a series of aromatic hydrocarbon radicals. These, in turn, have been used to calculate graph-theoretically the HOMO energies of a series of methylbenzenes in a hyperconjugative model. From a correlation of the calculated energies with the charge-transfer bands of their π-type molecular complexes, reasonable values of the hyperconjugative Hückel parameters of the methyl group have been found. The convenience resulting from the regressive use of FT's has been focussed on.

Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal’s triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP’s. Thus it has been shown that the entire eigenspectrum of ann-ring LP is included in that of (2n + 1)-ring LP. Correspondence between the eigenspectra of linear chains and LP’s is brought out by a recently developed vertex-alternation and squaring algorithm.