The Electronic Journal of Combinatorics· 2026Q1
Asymptotics for Graphically Divergent Series: Dense Digraphs and 2-SAT Formulae
- 2citations
- Q1SCImago
- 2026year
Short summary
A new method systematically derives complete asymptotic expansions for counting graph families in the dense regime by encoding coefficients into a bivariate 'coefficient generating function'.
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Key points
- Proposes a new method for systematic asymptotic expansion derivation for counting graph families.
- Encodes expansion coefficients into a bivariate 'coefficient generating function'.
- Reveals general properties of coefficient generating functions for closed-form asymptotics.
- Applications include connected graphs, irreducible tournaments, strongly connected digraphs, and 2-SAT formulae.
- Method accommodates structural variations like component counts, weights, and marked variables.
AI-generated from the title and abstract; the full text is not read.
Abstract
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of expansion coefficients into a special bivariate generating function, which we call a coefficient generating function. We show that coefficient generating functions possess certain general properties that make it possible to express asymptotics in a short closed form. Also, in most scenarios, we indicate a combinatorial meaning of the involved coefficients. Applications of our method include asymptotics of connected graphs, irreducible tournaments, strongly connected digraphs, 2-SAT formulae and contradictory strongly connected implication digraphs. Moreover, due to its flexibility, the method allows to treat a wide range of structural variations, including fixing the numbers of connected, irreducible, strongly connected and contradictory components, as well as source-like, sink-like and isolated ones, or adding weights and marking variables.
The authors' abstract, as published at the source. The Electronic Journal of Combinatorics, 2026 · DOI ↗
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Field: Physical and Theoretical Chemistry
Physical and Theoretical ChemistryChemistry