Combinatorics and Graph Theory

New Classifications of Labeled Rooted Growing Trees and Their Application to Simplifying of the Tree Representation of Functions

Authors: G. I. Kalmykov
Comments: Pages.

The article deals with labeled rooted growing trees. Research in this area, carried outby the author of this article over the past 35 years, has led to the creation of the conceptof tree classification of labeled graphs. This concept is the mathematical basis of the treesum method aimed at simplifying the representations of the coefficients of power series in classical statistical mechanics. This method was used to obtain tree representations of Mayer coefficients of expansions of pressure and density in terms of activity degrees, which are free from asymptotic catastrophe. The same method was used to obtain tree representations of thecoefficients of the expansion of the ratio of activity to density in terms of activity degrees.All these representations for n ≥ 7 are much simpler than the comparable Ree-Hooverrepresentations according to the complexity criteria defined on these representations. Treerepresentations of the coefficients of the expansion of the m-particles distribution functioninto a series in terms of activity degrees were also obtained. All the above representations ofthe coefficients of power series obtained by the trees sum method are free from the asymptoticcatastrophe.In order to provide a mathematical basis for constructing new, even less complexrepresentations of the coefficients of these power series, further development of the conceptof tree classification of labeled graphs was required.As part of solving the problem of further development of this concept, the article proposesnew classifications of labeled rooted growing trees. And on their basis, the theorem wasformulated and proved, which is the basis for simplifying the tree representations of functions,that is, its representations as a sum of labeled by trees products of functions.
Category: Combinatorics and Graph Theory