Structure of the flow and Yamada polynomials of cubic graphs

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.Comment: 22 page

On Zero-free Intervals of Flow Polynomials

This article studies real roots of the flow polynomial $F(G,\lambda)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $\xi_k$ be the supremum in $(1,2]$ such that $F(G,\lambda)$ has no real roots in $(1,\xi_k)$ for all graphs $G$ with $|W(G)|\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $\xi_k$ can be determined by considering a finite set of graphs and show that $\xi_k=2$ for $k\le 2$, $\xi_3=1.430\cdots$, $\xi_4=1.361\cdots$ and $\xi_5=1.317\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,\lambda)$ are real but some of these roots are not in the set $\{1,2,3\}$, then $|E|\ge |V|+17$ and $F(G,\lambda)$ has at least 9 real roots in $(1,2)$.Comment: 26 pages, 7 figure

On the imaginary parts of chromatic root

While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (that is, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We also show that for any fixed $p \in (0,1)$, almost every random graph $G$ in the Erd\"os-R\'enyi model has a non-real root.Comment: 4 figure

Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of Proposition 4.1, and adds related discussion. To appear in Combinatorics, Probability & Computin