Graph factorization: A new mode of application of a vertex‐alternation scheme

Abstract

AbstractLinear chains where the vertex weights change sign alternantly but are equal in magnitude were able to be reduced to smaller chains by a procedure analogous to that given by Coulson and Rushbrooke. The algorithm for constructing the reduced chains has been stated and proved. The results have been utilized, in conjunction with McClelland's graph‐factorization method using reflection (σ) planes, to reduce the HMO secular determinants of some chemical graphs to an extent beyond the ability of group theory. McClelland's σ‐plane algorithm, used repetitively where possible, produces factors whose sizes (nM) are equal to those (nG) of the group‐theoretic factor blocks. For linear polyacenes (LP), however, a new observation has been made: If the LP has an even number of fused rings, nM = nG; but when the LP has an odd number of fused rings, McClelland's process is effective in further reduction, i.e., nM < nG. In any case, however, the vertex alternation procedure reported in the present paper brings about further reduction. To demonstrate the utility of the present method, a sample calculation of the LUMO eigenvector graph theoretically has been shown for p‐benzoquinone and the result has been utilized to obtain an inductive effect HMO parameter of the methyl group from the charge‐transfer bands of some molecular complexes of methylated p‐benzoquinones.© 1993 John Wiley & Sons, Inc.