High-ordered spectral characterization of unicyclic graphs

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra

Building generalized linear models with ultrahigh dimensional features: A sequentially conditional approach

Conditional screening approaches have emerged as a powerful alternative to the commonly used marginal screening, as they can identify marginally weak but conditionally important variables. However, most existing conditional screening methods need to fix the initial conditioning set, which may determine the ultimately selected variables. If the conditioning set is not properly chosen, the methods may produce false negatives and positives. Moreover, screening approaches typically need to involve tuning parameters and extra modeling steps in order to reach a final model. We propose a sequential conditioning approach by dynamically updating the conditioning set with an iterative selection process. We provide its theoretical properties under the framework of generalized linear models. Powered by an extended Bayesian information criterion as the stopping rule, the method will lead to a final model without the need to choose tuning parameters or threshold parameters. The practical utility of the proposed method is examined via extensive simulations and analysis of a real clinical study on predicting multiple myeloma patients’ response to treatment based on their genomic profiles.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154458/1/biom13122-sup-0003-supmat.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154458/2/biom13122_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154458/3/biom13122.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154458/4/biom13122-sup-0002-Supplementary-072219.pd

Non-Abelian Quantum Hall Effect in Topological Flat Bands

Inspired by recent theoretical discovery of robust fractional topological phases without a magnetic field, we search for the non-Abelian quantum Hall effect (NA-QHE) in lattice models with topological flat bands (TFBs). Through extensive numerical studies on the Haldane model with three-body hard-core bosons loaded into a TFB, we find convincing numerical evidence of a stable $\nu=1$ bosonic NA-QHE, with the characteristic three-fold quasi-degeneracy of ground states on a torus, a quantized Chern number, and a robust spectrum gap. Moreover, the spectrum for two-quasihole states also shows a finite energy gap, with the number of states in the lower energy sector satisfying the same counting rule as the Moore-Read Pfaffian state.Comment: 5 pages, 7 figure