Quesitions of enumeration of selected classes of oriented and non-oriented trees and forests

E. I. Deza

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 111-128.

Résumé

In this article we consider questions of graph enumeration for some graphs of a special form. In fact, a number of new results have been proved on the number of spanning trees and spanning forests of graphs that play an important role in the applied problems of Information Theory. On the one hand, the properties of the spanning converging forests of oriented graphs involved in the construction of the mean first passage time quasi-metric, a generalized metric structure closely related to ergodic homogeneous Markov chains, are considered. On the other hand, the characteristics of spanning rooted forests and spanning converging forests of non-oriented and oriented graphs needed for the construction of a matrix of relative connectivity via forests, one of the measures of proximity of the vertices of graph structures, which plays an important role in solving of applied problems, have been studied. The consideration is based on a simple graph model, so-called caterpillar, and its oriented analogues. The other simple graph models, including oriented and non-oriented simple cycles and simple paths, where considered before. The first section (introduction) presents the history of the problem and provides an overview of the main ideas and results presented in the article. The role of graph models in the presentation and study of ergodic homogeneous Markov chains is considered. The matrix of relative connectivity via forests for non-oriented and oriented graphs is defined; its role for solving important applied problems of Information Theory is disclosed. The second section contains the basic definitions of Graph Theory necessary to formulate and prove the main results of the article. The definitions of a graph and an oriented graph, a spanning subgraph, a spanning rooted forest (for non-oriented graphs) and a spanning converging forest (for oriented graphs) are given. Some examples are represented (simple path, simple cycle, caterpillar and their oriented analogues). In the same section a number of properties of Fibonacci numbers necessary to obtain the main results of the article for the undirected case is formulated. In the third section, two theorems on the enumeration of graphs related to the construction of the mean first passage time matrix for a homogeneous ergodic Markov chain are proved. In fact, the number of spanning converging trees for the oriented caterpillar and the number of spanning rooted trees for the non-oriented caterpillar are given; the spanning forests consisting of two trees for the same graph structures are counted. Results for the oriented case are formulated in terms of values $2^k$, $k\geq 0$; results for the non-oriented case are formulated in terms of Fibonacci numbers $u_k$, $k\geq 1$. The proofs are based on elementary methods of enumerating Combinatorics. The fourth section presents the results related to enumeration of spanning forests needed for construction of the matrix of relative connectivity via forests for the non-oriented caterpillar and its oriented analogue. Total number of spanning converging forests (for oriented caterpillar) and total number of spanning rooted forests (for non-oriented caterpillar) are found; enumeration of the spanning converging forests, in which a vertex $i$ belongs to a tree converging to a vertex $j$ (for the oriented caterpillar), and enumeration of the spanning rooted forests, in which a vertex $i$ belongs to a tree with a root $j$ (for the non-oriented caterpillar) are represented. As before, results for the oriented case are formulated in terms of values $2^k$, $ k\geq 0$; results for the non-oriented case are formulated in terms of Fibonacci numbers $u_k$, $ k\geq 1$. The fifth section contains the examples of the the matrix of relative connectivity via forests. The sixth section (conclusion) presents the main conclusions of the article, outlines the ideas of further studies.

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