Séminaire Lotharingien de Combinatoire, 93B.116 (2025), 12 pp.
Michael Borinsky, Chiara Meroni and Maximilian Wiesmann
Asymptotic Count of Edge-Bicolored Graphs
Abstract.
We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs.
Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures. The asymptotic behavior is governed by the critical points of a polynomial.
As an application, we discuss the Ising model on a random 4-regular graph and show how its phase transitions arise from our formula.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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