Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank
Abstract
We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter f with f(∅)=1 has exponentially bounded edge-connection rank if and only if it is a mixed partition function. The bound is exact: for a real number R≥ 1, the connection ranks satisfy rk Mf,t≤ Rt for all t≥ 0 if and only if f has a model on a super vector space ℂk|2ℓ with k+2ℓ≤ R. Consequently the base of exponential growth of the connection ranks is the least number of colours of a model, the two dimensions of a minimal model are determined by f, and the parameters with a model on a prescribed ℂk|2ℓ are characterised. The proof organises fragments modulo the connection kernel into a rigid symmetric tensor category whose morphism spaces have the connection ranks as dimensions; the rank hypothesis and an argument of Schrijver make its additive idempotent completion semisimple, Deligne's theorem provides a fibre functor to super vector spaces, and the resulting super tensor network is identified with the Regts-Sevenster model exactly, circuit signs included. An appendix shows that in a rigid symmetric ℂ-linear category with End(1)=ℂ, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.
Related
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BibTeX
@misc{whistler2026mixed,
title = {{Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank}},
author = {William Whistler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.27198},
eprint = {2607.27198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27198},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank
AU - William Whistler
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2607.27198
UR - https://arxiv.org/abs/2607.27198
AB - We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter f with f(∅)=1 has exponentially bounded edge-connection rank if and only if it is a mixed partition function. The bound is exact: for a real number R≥ 1, the connection ranks satisfy rk M_f,t≤ R^t for all t≥ 0 if and only if f has a model on a super vector space ℂ^k|2ℓ with k+2ℓ≤ R. Consequently the base of exponential growth of the connection ranks is the least number of colours of a model, the two dimensions of a minimal model are determined by f, and the parameters with a model on a prescribed ℂ^k|2ℓ are characterised. The proof organises fragments modulo the connection kernel into a rigid symmetric tensor category whose morphism spaces have the connection ranks as dimensions; the rank hypothesis and an argument of Schrijver make its additive idempotent completion semisimple, Deligne's theorem provides a fibre functor to super vector spaces, and the resulting super tensor network is identified with the Regts-Sevenster model exactly, circuit signs included. An appendix shows that in a rigid symmetric ℂ-linear category with End(1)=ℂ, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.
ER -