20 research outputs found
Ricci-flat graphs with girth at least five
A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges.
Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau,
Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009].
In this paper, we classified all Ricci-flat connected graphs with girth at
least five: they are the infinite path, cycle (), the
dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We
also construct many Ricci-flat graphs with girth 3 or 4 by using the root
systems of simple Lie algebras.Comment: 14 pages, 15 figure
Discrete trace function and Poincaré inequality for the study of some linear systems
Postprint (published version
Existence and nonexistence of minimizer for Thomas-Fermi-Dirac-Von Weizs\"{a}cker model on lattice graph
The focus of our paper is to investigate the possibility of a minimizer for
the Thomas-Fermi-Dirac-von Weizs\"{a}cker model on the lattice graph
. The model is described by the following functional:
\begin{equation*}
E(\varphi)=\sum_{y\in\mathbb{Z}^{3}}\left(|\nabla\varphi(y)|^2+
(\varphi(y))^{\frac{10}{3}}-(\varphi(y))^{\frac{8}{3}}\right)+
\sum_{x,y\in\mathbb{Z}^{3}\atop ~\ y\neq
x\hfill}\frac{{\varphi}^2(x){\varphi}^2(y)}{|x-y|}, \end{equation*} with the
additional constraint that
is sufficiently small. We also prove the nonexistence of a minimizer provided
the mass is adequately large. Furthermore, we extend our analysis to a
subset and prove the nonexistence of a
minimizer for the following functional: \begin{equation*}
E(\Omega)=|\partial\Omega|+\sum_{x,y\in\Omega\atop ~y\neq
x\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that
is sufficiently large.Comment: 21 page