Evaluations of Tutte polynomials of regular graphs
Ferenc Bencs, P\'eter Csikv\'ari

TL;DR
This paper investigates the asymptotic behavior of Tutte polynomials for sequences of regular graphs with increasing girth, revealing a universal limit depending only on the degree and parameters, and extends previous results to broader cases.
Contribution
It establishes a general limit theorem for Tutte polynomials of large-girth regular graphs, unifying and extending prior specific results for spanning trees and forests.
Findings
Limit of normalized Tutte polynomials exists for large-girth regular graphs.
The limit is independent of the parameter y within [0,1].
Results apply almost surely to random regular graphs.
Abstract
Let be the Tutte polynomial of a graph . In this paper we show that if is a sequence of -regular graphs with girth , then for and we have where independently of if . If is a sequence of random -regular graphs, then the same statement holds true asymptotically almost surely. This theorem generalizes results of McKay (, spanning trees of random -regular graphs) and Lyons (, spanning trees of large-girth -regular graphs). Interesting special cases are counting the number of spanning forests, …
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsStochastic processes and statistical mechanics · Graph theory and applications · Advanced Combinatorial Mathematics
