Chromatic Zeros on the Limit $G^{(p,\ell)}_\infty$ of the Family $G^{(p,\ell)}_m$ of Hierarchical Graphs

TL;DR

This paper analyzes the zeros of the chromatic polynomial for a family of hierarchical graphs using a renormalization group approach, revealing complex behavior and critical points in the complex plane.

Contribution

It extends previous work on chromatic zeros to higher parameters, providing explicit calculations and insights into the asymptotic behavior of these zeros for a broad class of graphs.

Findings

Calculated the accumulation set of chromatic zeros in the complex q-plane.

Identified the maximal crossing point q_c and other crossing points of the locus ${\cal B}_q$.

Extended analysis to higher p and l values beyond the (2,2) case.

Abstract

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained from $G^{(p,\ell)}_{m-1}$ by replacing each edge (i.e., bond) on $G^{(p,\ell)}_m$ by $p$ paths each of length $\ell$ edges, starting with the tree graph $T_2$. Our method uses the property that the chromatic polynomial $P(G,q)$ of a graph $G$ is equal to the $v=-1$ evaluation of the partition function of the $q$-state Potts model, together with (i) the property that $Z(G^{(p,\ell)}_m,q,v)$ can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of $Z(G^{(p,\ell)}_{m-1},q,v')$, where $v'=F_{(p,\ell),q}(v)$ is a rational function of $v$ and $q$ and (ii) ${\cal B}_q(p,\ell)(v)$ is the locus in the complex $q$-plane that separates regions of different asymptotic behavior of the $m$-fold iterated RG transformation $F_{(p,\ell),q}(v)$ in the $m \to \infty$ limit. Thus, our results involve calculations of region diagrams in the complex $q$-plane showing the type of behavior that occurs in the $m \to \infty$ limit of the $m$-fold iterated RG transformation mapping $F_{(p,\ell),q}(v)$ starting with the initial value $v=v_0=-1$. Calculations are presented of the maximal point $q_c(G^{(p,\ell)}_\infty)$ at which the locus ${\cal B}_q$ crosses the real-$q$ axis, as well as several other points at which, depending on $p$ and $\ell$, the locus ${\cal B}_q$ crosses this axis. We give explicit results for a variety of $(p,\ell)$ cases and observe a number of interesting features. Calculations of the ground-state degeneracy of the Potts antiferromagnet at $q_c(G^{(p,\ell)}_\infty)$ are presented. This work extends a previous study with R. Roeder of the $(p,\ell)=(2,2)$ case to higher $p$ and $\ell$ values.