相关论文: Fire retainment on Cayley graphs
The extended persistence diagram introduced by Cohen-Steiner, Edelsbrunner, and Harer is an invariant of real-valued continuous functions, which are $\mathbb{F}$-tame in the sense that all open interlevel sets have degree-wise…
We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected…
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by…
We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\trianglelefteq G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not…
We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde…
In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing…
Kim, K\"uhn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi by proving a `blow-up lemma for approximate decompositions' which states…
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup…
Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these…
The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent…
In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the…
We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen, and…
Preservation theorems provide a direct correspondence between the syntactic structure of first-order sentences and the closure properties of their respective classes of models. A line of work has explored preservation theorems relativised…
The non extensive statistics proposed by C. Tsallis has found wide applicability, being present even in the description of experimental data from high energy collisions. A system with a fractal structure in its energy-momentum space, named…
We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(\epsilon,\delta)$-domains $\Omega\subset\mathbb{R}^n$ with a $d$-set…
One of the prominent areas of research in graph theory is the degree-diameter problem, in which we seek to determine how many vertices a graph may have when constrained to a given degree and diameter. Different variants of this problem are…
We show that the geometric deformation of shearing yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in an unbounded strip. The proof is based on the Hardy inequality due to the shearing…