On the second largest eigenvalue of the signless Laplacian

Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G)$ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality $q_{k}(G) =n-2,$ where $1<k\leq n.$ In particular, this result solves an open problem raised by Wang, Belardo, Huang and Borovicanin. We also show that [ q_{2}(G) \geq\delta(G)] and determine that equality holds if and only if $G$ is one of the following graphs: a star, a complete regular multipartite graph, the graph $K_{1,3,3},$ or a complete multipartite graph of the type $K_{1,...,1,2,...,2}$.Comment: This version fills a gap in one proof, noticed by Rundan Xin

Modulated phases and devil's staircases in a layered mean-field version of the ANNNI model

We investigate the phase diagram of a spin-$1/2$ Ising model on a cubic lattice, with competing interactions between nearest and next-nearest neighbors along an axial direction, and fully connected spins on the sites of each perpendicular layer. The problem is formulated in terms of a set of noninteracting Ising chains in a position-dependent field. At low temperatures, as in the standard mean-feild version of the Axial-Next-Nearest-Neighbor Ising (ANNNI) model, there are many distinct spatially commensurate phases that spring from a multiphase point of infinitely degenerate ground states. As temperature increases, we confirm the existence of a branching mechanism associated with the onset of higher-order commensurate phases. We check that the ferromagnetic phase undergoes a first-order transition to the modulated phases. Depending on a parameter of competition, the wave number of the striped patterns locks in rational values, giving rise to a devil's staircase. We numerically calculate the Hausdorff dimension $D_{0}$ associated with these fractal structures, and show that $D_{0}$ increases with temperature but seems to reach a limiting value smaller than $D_{0}=1$.Comment: 17 pages, 6 figure

Greenberger-Horne-Zeilinger state generation with linear optical elements

We propose a scheme to probabilistically generate Greenberger-Horne-Zeilinger (GHZ) states encoded on the path degree of freedom of three photons. These photons are totally independent from each other, having no direct interaction during the whole evolution of the protocol, which remarkably requires only linear optical devices to work, and two extra ancillary photons to mediate the correlation. The efficacy of the method, which has potential application in distribited quantum computation and multiparty quantum communication, is analyzed in comparison with similar proposals reported in the recent literature. We also discuss the main error sources that limit the efficiency of the protocol in a real experiment and some interesting aspects about the mediator photons in connection with the concept of spatial nonlocality