Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Eigenvalue problems involving the Laplace operator on bounded domains lead to a discrete or a continuous set of eigenvalues. In this paper we highlight the case of an eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that proble...

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 mixing time of an ergodic, reversible Markov chain can be bounded in terms of the eigenvalues of the chain: specifically, the second-largest eigenvalue and the smallest eigenvalue. It has become standard to focus only on the secondlargest eigenvalue, by making the Markov chain “lazy”. (A lazy chain does nothing at each step with probability at least 1 2 , and has only nonnegative eigenvalue...

We give several new formulas of the least positive Fredholm eigenvalue for a quasicircle, answering a problem posed recently by Kühnau. During the proof, we show that the Grunsky functionals corresponding to the two complementary domains of the quasi-circle are the same and equal to the reciprocal of the Fredholm eigenvalue.

We consider a second order three point boundary value problem for dynamic equations on time scales and establish criteria for the existence of at least two positive solutions of an eigenvalue problem by an application of a fixed point theorem in cones. Existence result for non-eigenvalue problem is also given by the monotone method.