5 research outputs found
Connected Hypergraphs with Small Spectral Radius
In 1970 Smith classified all connected graphs with the spectral radius at
most . Here the spectral radius of a graph is the largest eigenvalue of its
adjacency matrix. Recently, the definition of spectral radius has been extended
to -uniform hypergraphs. In this paper, we generalize the Smith's theorem to
-uniform hypergraphs. We show that the smallest limit point of the spectral
radii of connected -uniform hypergraphs is . We
discovered a novel method for computing the spectral radius of hypergraphs, and
classified all connected -uniform hypergraphs with spectral radius at most
.Comment: 20 pages, fixed a missing class in theorem 2 and other small typo
Existence results for some nonlinear elliptic systems on graphs
In this paper, several nonlinear elliptic systems are investigated on graphs.
One type of the sobolev embedding theorem and a new version of the strong
maximum principle are established. Then, by using the variational method, the
existence of different types of solutions to some elliptic systems is
confirmed. Such problems extend the existence results on closed Riemann surface
to graphs and extend the existence results for one single equation on graphs
[A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear
elliptic systems on graphs. Such problems can also be viewed as one type of
discrete version of the elliptic systems on Euclidean space and Riemannian
manifold
Gradient estimates for the weighted porous medium equation on graphs
In this paper, we study the gradient estimates for the positive solutions of
the weighted porous medium equation
on graphs for , which is a nonlinear version of the heat equation.
Moreover, as applications, we derive a Harnack inequality and the estimates of
the porous medium kernel on graphs. The obtained results extend the results of
Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9]