Enumeration of tree-type diagrams assembled from oriented chains of edges
O. Khorunzhiy

TL;DR
This paper derives explicit formulas for counting and summing weighted tree-type diagrams constructed from oriented chains, relevant to cumulant expansions in Erdős-Rényi random matrix models, using a modified Prüfer code.
Contribution
It introduces a novel application of a modified Prüfer code to explicitly enumerate and sum weighted tree diagrams assembled from oriented chains.
Findings
Explicit enumeration formulas for tree-type diagrams from oriented chains
Closed-form expressions for weighted sums of diagrams with edge multiplicities
Extension of results to diagrams assembled from non-regular chains
Abstract
We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-R\'enyi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type diagrams assembled from oriented chains of edges. Using this modified Pr\"ufer codification, we get an explicit expression for sum overs weighted tree-type diagrams with a weight depending on multiplicity of edges. We describe similar results for tree-type diagrams assembled from chains that are not necessarily regular.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Theoretical and Computational Physics · semigroups and automata theory
