相关论文: Bounds on the Complex Zeros of (Di)Chromatic Polyn…

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex…

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known…

I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their…

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $\chi_G(x)$, lie inside the disk centered at $0$ of radius $4.25 \Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. This improves on a recent…

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(z)$ (in $\mathbb{C}$) lie outside the disk of radius $5.02 \Delta$ centered at $0$. This improves on the previously best known…

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$ C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]} $$ This…

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the…

Let $P(G,q)$ be the chromatic polynomial for coloring the $n$-vertex graph $G$ with $q$ colors, and define $W=\lim_{n \to \infty}P(G,q)^{1/n}$. Besides their mathematical interest, these functions are important in statistical physics. We…

We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs $K_b$ with $b=5,6$…

The purpose of the present paper is to provide, for all pairs of integers $(\Delta,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(\Delta, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $\Delta$ and…

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) Z_G(q,w) of a graph G with general complex edge weights w = {w_e}.…

The zero-temperature $q$-state Potts model partition function for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $P(G,q)=\sum_{j=1}^{N_{G,\lambda}}c_{G,j}(\lambda_{G,j})^{L_x}$, and is equivalent to the…

We present exact calculations of the zero-temperature partition function (chromatic polynomial) and the (exponent of the) ground-state entropy $S_0$ for the $q$-state Potts antiferromagnet on families of cyclic and twisted cyclic (M\"obius)…

We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set…

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, showing density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)>3/2$ and we…

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…

We present exact solutions for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial $P$) on tube sections of the simple cubic lattice of fixed transverse size $L_x \times…

We study the chromatic polynomial P_G(q) for m \times n triangular-lattice strips of widths m <= 12_P, 9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary…

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m \times n rectangular subsets of the square lattice, with m \le 8 (free or periodic transverse boundary conditions) and n…

The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these…