THE FACTORISATION OF CHEMICAL GRAPHS AND THEIR POLYNOMIALS: A SYSTEMATIC STUDY OF CERTAIN TREES

Characteristic polynomials of the set of 284 trees which have 1-12 vertices of valency 1-3 have been examined for possible factors (divisors) using polynomial division. Twenty of these trees are prime in the sense that they contain no other trees in the set as factors. The remaining 264 trees can all be constructed from a subset of 5 trees and a set of 152 non-graphical polynomials. Some of these polynomials exhibit iso- or sub-spectral relationships with acyclic (matching) polynomials of certain cyclic structures. A few cyclic factors of trees are noted briefly. Twenty pairs and one triad of the trees examined are isospectral.