Potts internal-energy bounds for regular graphs

Let $G$ be a $d$-regular graph, let $q\ge 2$, and let $\beta>0$. Write $K_{d,d}$ for the complete $d$-regular bipartite graph and $K_{d+1}$ for the complete graph on $d+1$ vertices. The internal energy and free energy per particle are denoted by $U^q_G(\beta)$ and $F^q_G(\beta)$, respectively.

Regular-graph Potts bounds. The conjectured inequalities are

$$U^q_{K_{d,d}}(\beta)\le U^q_G(\beta)\le U^q_{K_{d+1}}(\beta),$$

and, in particular,

$$F^q_{K_{d+1}}(\beta)\le F^q_G(\beta)\le F^q_{K_{d,d}}(\beta).$$

These bounds extend the theorem proved in the paper for cubic graphs; the case $d=2$ is stated to follow by direct calculation, while the higher-regularity case remains conjectural.