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.
From: Khaydar Nurligareev [view email]
[v1]
Sun, 8 Oct 2023 21:10:56 UTC (63 KB)
[v2]
Fri, 29 Nov 2024 17:16:50 UTC (75 KB)
[v3]
Wed, 5 Aug 2026 19:46:07 UTC (59 KB)
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