Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials

TL;DR

This paper establishes new homomorphism inequalities for antiferromagnetic graphs using Lorentzian polynomials, advancing understanding of graph parameters like independent sets and colorings, and making progress on conjectures in graph theory and statistical physics.

Contribution

It introduces novel homomorphism inequalities for antiferromagnetic graphs, leveraging Lorentzian polynomial theory, and advances conjectures related to graph colorings and extremal graph counts.

Findings

Proves inequalities for homomorphisms involving blow-ups of bipartite graphs.

Extends results to complete multipartite graphs, paths, and cycles.

Progresses towards Zhao's conjecture on q-colorings.

Abstract

An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex-colourings, as well as their relaxations in statistical physics. We obtain homomorphism inequalities for various graphs $H$ and antiferromagnetic graphs~$G$ of the form \[ \lvert\operatorname{Hom}(H,G)\rvert^2 \leq \lvert\operatorname{Hom}(H\times K_2,G)\rvert, \] where $H\times K_2$ denotes the tensor product of $H$ and $K_2$. Firstly, we show that the inequality holds for any $H$ obtained by blowing up vertices of a bipartite graph into complete graphs and any antiferromagnetic $G$. In particular, one can take $H=K_{d+1}$, which already implies a new result for the Sah--Sawhney--Stoner--Zhao conjecture on the maximum number of $d$-regular graphs in antiferromagnetic graphs. Secondly, the inequality also holds for $G=K_q$ and those $H$ obtained by blowing up vertices of a bipartite graph into complete multipartite graphs, paths or even cycles. Both results can be seen as the first progress towards Zhao's conjecture on $q$-colourings, which states that the inequality holds for any $H$ and $G=K_q$, after his own work. Our method leverages on the emerging theory of Lorentzian polynomials due to Br\"and\'en and Huh and log-concavity of the list colourings of bipartite graphs, which may be of independent interest.