In this paper we prove an isoperimetric inequality for the twisted Dirichlet eigenvalue which was introduced by Barbosa and Bérard in the context of constant mean curvature surfaces. More precisely, we show that in the Euclidean case this eigenvalue is minimized by the union of two equal balls.

We develop a model of regular, infinite hypertrees, to mimic for hypergraphs what infinite trees do for graphs. We then examine two notions of spectra or “first eigenvalue” for the infinite tree, obtaining a precise value for the first notion and obtaining some estimates for the second. The results indicate agreement of the first eigenvalue of the infinite hypertree with the “second eigenvalue”...

The class of nonsingular connected weighted directed graphs with an unweighted undirected branch is considered in this article. This paper investigates the monotonicity properties of the first eigenvectors of such graphs along certain paths. The paper describes how the first eigenvalue of such graphs changes under some perturbation. It is shown that replacing a branch which is a tree by a path ...