A characterization and an application of weight-regular partitions of graphs

A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $\nu_u$ of the Perron eigenvector $\mathbf{\nu}$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity by using a Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular graphs. In addition, we show an application of weight-regular partitions to study graphs that attain equality in the classical Hoffman's lower bound for the chromatic number of a graph, and we show that weight-regularity provides a condition under which Hoffman's bound can be improved

Tunneling for spatially cut-off P(ϕ)2P(\phi)_2-Hamiltonians

We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off $P(\phi)_2$-Hamiltonian under semi-classical limit. The corresponding classical equation of the $P(\phi)_2$-field is a nonlinear Klein-Gordon equation which is an infinite dimensional Newton's equation. We determine the semi-classical limit of the lowest eigenvalue of the spatially cut-off $P(\phi)_2$-Hamiltonian in terms of the Hessian of the potential function of the Klein-Gordon equation. Moreover, we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast under semi-classical limit when the potential function is double well type. In fact, we prove that the exponential decay rate is greater than or equal to the Agmon distance between two zero points of the symmetric double well potential function. The Agmon distance is a Riemannian distance on the Sobolev space H^{1/2}(\RR) defined by a Riemannian metric which is formally conformal to $L^2$-metric. Also we study basic properties of the Agmon distance and instanton.Comment: 1. I proved that the exponential decay rate of the gap of spectrum is greater than or equal to the Agmon distance between zero points in the case of spatially cut-off P(\phi)_2-Hamiltonians. 2. I proved the existence of geodesic for the Agmon metric and also the existence of instanto

Molière's life and a critical study of some of his famous works

Thesis (M.A.)--Boston Universit

The Rise of Latino Protestants

Excerpt: During one of my first visits to a church in San Antonio for the Latino Protestant Congregations Project, the pastor invited a church member to speak about his experience in a federal immigration detention center. An elderly gentleman rose from his seat with a Bible tucked under his arm. For the next hour, this man, a Salvadoran undocumented immigrant, told his story

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed $L_{p}$ spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering