Extremal f-trees and embedding spaces for molecular graphs

Abstract

Problems concerning embedding trees in lattice-graph or Euclidean spaces are considered. A tree is defined to be ‘almost-embeddable’ in a lattice-graph if a sequence derived from the distance degree sequence of the lattice-graph and a corresponding sequence for the tree satisfy a specified inequality. This inequality is such that every tree that is embeddable in the lattice-graph is in the set of almost-embeddable trees. For Euclidean space embeddings the lattice-graph sequence is replaced by a sequence defined in terms of sphere packing numbers. This work has two practical objectives: Firstly, to furnish a framework within which intuitive chemical and physical notions about embedding spaces can be made explicit and self-consistent. Secondly, to obtain useable criteria which will exclude from statistical mechanical averaging procedures those molecular species which are inconsistent with a postulated embedding space. The inequality proposed here meets these objectives for molecular trees and its implications for chemical and physical theory are discussed in some detail.