Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called K\"onig-Egerv\'ary if the sum of its independence number ($\alpha (G)$) and matching number ($\mu (G)$) equals $|V(G)|$. It is known that bipartite graphs are K\"onig-Egerv\'ary. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2\alpha (G)$ (proved by Short in 2016). A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha (G) - \mu (G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.Comment: 18 pages, 2 figure

Can LIGO Detect Asymmetric Dark Matter?

Dark matter from the galactic halo can accumulate in neutron stars and transmute them into sub-2.5 $M_\odot$ black holes if the dark matter particles are heavy, stable, and have interactions with nucleons. We show that non-detection of gravitational waves from mergers of such low-mass black holes can constrain the interactions of asymmetric dark matter particles with nucleons. We find benchmark constraints with LIGO O3 data, viz., $\sigma_{\chi n} \geq {\cal O}(10^{-47})$ cm$^2$ for bosonic DM with $m_\chi\sim$ PeV (or $m_\chi\sim$ GeV, if they can Bose-condense) and $\geq {\cal O}(10^{-46})$ cm$^2$ for fermionic DM with $m_\chi \sim 10^3$ PeV. These bounds depend on the priors on DM parameters and on the currently uncertain binary neutron star merger rate density. However, if null-detection continues with increased exposure over the next decade, LIGO will set remarkable constraints. We find the forecasted sensitivity to heavy asymmetric dark matter to be world-leading, viz., dipping many orders of magnitude below the neutrino floor and completely testing the dark matter solution to missing pulsars in the Galactic center, and demonstrate a windfall science-case for gravitational wave detectors.Comment: 14 pages, 6 figures. Comments welcom

Machine learning approach to genome of two-dimensional materials with flat electronic bands

Many-body physics of electron-electron correlations plays a central role in condensed mater physics, it governs a wide range of phenomena, stretching from superconductivity to magnetism, and is behind numerous technological applications. To explore this rich interaction-driven physics, two-dimensional (2D) materials with flat electronic bands provide a natural playground thanks to their highly localised electrons. Currently, thousands of 2D materials with computed electronic bands are available in open science databases, awaiting such exploration. Here we used a new machine learning algorithm combining both supervised and unsupervised machine intelligence to automate the otherwise daunting task of materials search and classification, to build a genome of 2D materials hosting flat electronic bands. To this end, a feedforward artificial neural network was employed to identify 2D flat band materials, which were then classified by a bilayer unsupervised learning algorithm. Such a hybrid approach of exploring materials databases allowed us to reveal completely new material classes outside the known flat band paradigms, offering new systems for in-depth study on their electronic interactions

Impact of diabetes mellitus on ventricular structure, arterial stiffness, and pulsatile hemodynamics in heart failure with preserved ejection fraction

Background-Heterogeneity in the underlying processes that contribute to heart failure with preserved ejection fraction (HFpEF) is increasingly recognized. Diabetes mellitus is a frequent comorbidity in HFpEF, but its impact on left ventricular and arterial structure and function in HFpEF is unknown. Methods and Results-Weassessed the impact of diabetesmellitus on left ventricular cellular and interstitial hypertrophy (assessedwith cardiacmagnetic resonance imaging, including T1mapping pregadolinium and postgadolinium administration), arterial stiffness (assessed with arterial tonometry), and pulsatile arterial hemodynamics (assessed with in-office pressure-flow analyses and 24-hour ambulatory monitoring) among 53 subjects with HFpEF (32 diabetic and 21 nondiabetic subjects). Despite few differences in clinical characteristics, diabetic subjects with HFpEF exhibited a markedly greater left ventricular mass index (78.1 [95% CI, 70.4-85.9] g versus 63.6 [95% CI, 55.8-71.3] g; P=0.0093) and indexed extracellular volume (23.6 [95% CI, 21.2-26.1] mL/m(2) versus 16.2 [95% CI, 13.1-19.4] mL/m(2); P=0.0008). Pronounced aortic stiffening was also observed in the diabetic group (carotid-femoral pulse wave velocity, 11.86 [95% CI, 10.4-13.1] m/s versus 8.8 [95% CI, 7.5-10.1] m/s; P=0.0027), with an adverse pulsatile hemodynamic profile characterized by increased oscillatory power (315 [95% CI, 258-373] mWversus 190 [95% CI, 144-236] mW; P=0.0007), aortic characteristic impedance (0.154 [95% CI, 0.124-0.183] mmHg/mL per second versus 0.096 [95% CI, 0.072-0.121] mm Hg/mL per second; P=0.0024), and forward (59.5 [95% CI, 52.8-66.1] mm Hg versus 40.1 [95% CI, 31.6-48.6] mm Hg; P=0.0010) and backward (19.6 [95% CI, 16.2-22.9] mm Hg versus 14.1 [95% CI, 10.9-17.3] mm Hg; P=0.0169) wave amplitude. Abnormal pulsatile hemodynamics were also evident in 24-hour ambulatory monitoring, despite the absence of significant differences in 24-hour systolic blood pressure between the groups. Conclusions-Diabetes mellitus is a key determinant of left ventricular remodeling, arterial stiffness, adverse pulsatile hemodynamics, and ventricular-arterial interactions in HFpEF

Strain driven emergence of topological non-triviality in YPdBi thin films

Half-Heusler compounds exhibit a remarkable variety of emergent properties such as heavy-fermion behaviour, unconventional superconductivity and magnetism. Several of these compounds have been predicted to host topologically non-trivial electronic structures. Remarkably, recent theoretical studies have indicated the possibility to induce non-trivial topological surface states in an otherwise trivial half-Heusler system by strain engineering. Here, using magneto-transport measurements and first principles DFT-based simulations, we demonstrate topological surface states on strained [110] oriented thin films of YPdBi grown on (100) MgO. These topological surface states arise in an otherwise trivial semi-metal purely driven by strain. Furthermore, we observe the onset of superconductivity in these strained films highlighting the possibility of engineering a topological superconducting state. Our results demonstrate the critical role played by strain in engineering novel topological states in thin film systems for developing next-generation spintronic devices.Comment: 20 pages, 5 Figure