Related papers: Dimers, webs, and local systems
A well known theorem due to Kasteleyn states that the partition function of an Ising model on an arbitrary planar graph can be represented as the Pfaffian of a skew-symmetric matrix associated to the graph. This results both embodies the…
In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the…
We study the set of all determinants of adjacency matrices of graphs with a given number of vertices.
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V(G) \to N$ is a mapping. For a nonnegative…
The local metric dimension ${\rm dim}_l$ in relation to the clique number $\omega$ is investigated. It is proved that if $\omega(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the…
A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general…
Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not necessarily with the identity elements). A bijective map $\varphi$ of $\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative isomorphism} if {$\varphi…
We propose a sampling algorithm to perform system identification from a set of input-output graph signal pairs. The dynamics of the systems we study are given by a partially known adjacency matrix and a generic parametric graph filter of…
We initiate the study of computational problems on $k$-mers (strings of length $k$) in labeled graphs. As a starting point, we consider the problem of counting the number of distinct $k$-mers found on the walks of a graph. We establish that…
The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last…
This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we…
A biperiodic planar network is a pair $(G,c)$ where $G$ is a graph embedded on the torus and $c$ is a function from the edges of $G$ to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its…
The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of $k$-WL to recognition of graph properties. Let $G$ be an input graph with $n$…
Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs.…
The behaviour of many real-world phenomena can be modelled by nonlinear dynamical systems whereby a latent system state is observed through a filter. We are interested in interacting subsystems of this form, which we model by a set of…
We consider the problem of analyzing the heterogeneity of clustering distributions for multiple groups of observed data, each of which is indexed by a covariate value, and inferring global clusters arising from observations aggregated over…
The Kac-Ward formula allows to compute the Ising partition function on a planar graph G with straight edges from the determinant of a matrix of size 2N, where N denotes the number of edges of G. In this paper, we extend this formula to any…
For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings,…
The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be "computed" geometrically, in the way predicted by the "qantization commutes with reduction" principle…
Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the…