Related papers: The Poisson's Problems on graphs

We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above…

Let $L$ be a linear differential operator acting on functions defined over an open set $\mathcal{D}\subset \mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{x\in\mathcal{D}}$ whose sample…

The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{\omega_1, \dots , \omega_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,\omega_1), \dots , d(u,\omega_n))$. A vertex…

For discrete weighted graphs there is sufficient literature about the Cheeger cut and the Cheeger problem, but for metric graphs there are few results about these problems. Our aim is to study the Cheeger cut and the Cheeger problem in…

The degree/diameter problem is the problem of finding the largest possible number of vertices $n_{\Delta,D}$ in a graph of given degree $\Delta$ and diameter $D$. We consider the problem for the case of diameter $D=2$. William G Brown gave…

O'Donnell, Wright, Wu and Zhou [SODA 2014] introduced the notion of robustly asymmetric graphs. Roughly speaking, these are graphs in which for every $0 \le \rho \le 1$, every permutation that permutes a $\rho$ fraction of the vertices maps…

Consider the random graph $G({\mathcal P}_{n},r)$ whose vertex set ${\mathcal P}_{n}$ is a Poisson point process of intensity $n$ on $(- \frac{1}{2}, \frac{1}{2}]^d$, $d \geq 2$. Any two vertices $X_i,X_j \in {\mathcal P}_{n}$ are connected…

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit…

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…

In this paper, we study large and moderate deviation principles for stochastic partial differential equations (SPDEs) on metric graphs and their associated multiscale models via the weak convergence approach, providing a refined…

Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions defined on $V$. Let $\Delta_p$ be the discrete $p$-Laplacian on $G$ with $p>1$ and $L=\Delta_p-k$, where $k\in C(V)$ is positive everywhere. Consider the operator…

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some…

Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$…

Let $\Delta =\{ \delta_1,\delta_2,...,\delta_m \} $ be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function $D_{\Delta }(z,u_1,u_2,...,u_m),$ which counts polygon dissections and marks…

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed…

Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric…

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. This brings us to the following natural parameterized questions:…

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of…

We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ \Omega,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\…

Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a…