Symbolic method and directed graph enumeration
\'Elie de Panafieu, Sergey Dovgal
TL;DR
This paper introduces the arrow product, a new generating function technique for enumerating directed graphs, simplifying proofs of existing results and extending enumeration to graphs with specified strongly connected components.
Contribution
The paper presents the arrow product method, offering a systematic approach to directed graph enumeration and unifying previous results with shorter proofs.
Findings
Simplified proofs for the number of directed acyclic graphs
Unified enumeration of strongly connected directed graphs
Extended enumeration results for graphs with specified strongly connected components
Abstract
We introduce the arrow product, a systematic generating function technique for directed graph enumeration. It provides short proofs for previous results of Gessel on the number of directed acyclic graphs and of Liskovets, Robinson and Wright on the number of strongly connected directed graphs. We also recover Robinson's enumerative results on directed graphs where all strongly connected components belong to a given family.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Graph theory and applications · Markov Chains and Monte Carlo Methods