Related papers: Exact Potts/Tutte Polynomials for Hammock Chain Gr…
We report some new results on the complex-temperature (CT) singularities of $q$-state Potts models on the square lattice. We concentrate on the problematic region $Re(a) < 0$ (where $a=e^K$) in which CT zeros of the partition function are…
We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the linear…
The distinguishing result of this paper is a $\mathbf{P}$-time enumerable partition of all the potential perfect matchings in a bipartite graph. This partition is a set of equivalence classes induced by the missing edges in the potential…
This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that…
Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta> 1/2$. Let $\Psi(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic…
We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal…
We present exact calculations of the zero-temperature partition function, $Z(G,q,T=0)$, and ground-state degeneracy (per site), $W({G},q)$, for the $q$-state Potts antiferromagnet on a number of families of graphs ${G}$ for which the…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$,…
We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q>= 2 and gamma. (Relative to the classical (x,y)…
The Tutte polynomial is an important invariant of graphs and matroids. Chen and Guo \emph{[Adv. in Appl. Math. 166 (2025) 102868.]} proved that for a $(k+1)$-edge connected graph $G$ and for any $i$ with $0\leq i <\frac{3(k+1)}{2}$,…
A recent result has demonstrated that the Bethe partition function always lower bounds the true partition function of binary, log-supermodular graphical models. We demonstrate that these results can be extended to other interesting classes…
We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge…
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…
In this paper we consider the algorithmic problem of sampling from the Potts model and computing its partition function at low temperatures. Instead of directly working with spin configurations, we consider the equivalent problem of…
The Potts conformal field theory is an analytic continuation in the central charge of conformal field theory describing the critical two-dimensional $Q$-state Potts model. Four-point functions of the Potts conformal field theory are…
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…
In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class,…
This paper studies the problem of counting homomorphisms from a bipartite source graph to a bipartite target graph. An exact formula is first derived for the number of homomorphisms from a complete bipartite graph into a general bipartite…
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct…