Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials

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.