Related papers: A graph counterexample to davies' conjecture

Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically…

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.

We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type…

We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an ergodicity assumption. For the proof we use Davies' perturbation method, where we show a maximal…

We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck…

The paper deals with point-wise estimates for the heat kernel of a nonlocal convolution type operator with a kernel that decays at least exponentially at infinity. It is shown that the large time behaviour of the heat kernel depends…

In this paper I present a counter-example to the conjecture: The Fiedler vector for the graph Laplacian of a tree has its most extreme values at the verticies which are the farthest apart. This counter-example looks roughly like a flower…

In 2013 Andriy V. Bondarenko showed how to construct a two-distance counterexample to Borsuk's conjecture from any strongly regular graph whose vertex set is not the union of at most $f+1$ cliques (sets of pairwise adjacent vertices) where…

A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and…

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional…

Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ is a graph whose vertex set is the set of $r$-matchings in $G$ and two vertices are adjacent if their corresponding matchings are edge-disjoint. In…

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…

Let $(M,g)$ be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point $C$ is rotationally invariant, but not necessarily flat, we give an explicit…

A well-known high-school problem asking the final temperature of two spheres that are given the same amount of heat, one lying on a table and the other hanging from a thread, is re-examined. The conventional solution states that the sphere…

Using the graphs of prisms and Tutte Fragments, we construct an infinite family of hamiltonian and non-hamiltonian graphs in which Tutte's counterexample to Tait's conjecture appears in a certain sense as a minimal element. We observe that…

One of the most important characteristics of a quantum graph is the average density of resonances, $\rho = \frac{\mathcal{L}}{\pi}$, where $\mathcal{L}$ denotes the length of the graph. This is a very robust measure. It does not depend on…

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete…

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ \pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and…

For two graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality $n,$ the friends-and-strangers graph $\mathsf{FS}(X,Y)$ was recently defined by Defant and Kravitz. The vertices of $\mathsf{FS}(X,Y)$ are the bijections…

In this paper we present an explicit counterexample of degree $n=7$, which shows that the conjecture proposed by Li et al. \cite{Li2013} regarding the first derivative bounds for rational B\'ezier curves is generally false. We further…