Related papers: On the maximum angle between copositive matrices

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W*-subalgebras of an inclusion of W*-algebras with a normal conditional expectation with finite probabilistic index are…

We consider the Max-Cut problem. Let $G = (V,E)$ be a graph with adjacency matrix $(a_{ij})_{i,j=1}^{n}$. Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{\theta_1, \theta_2, \dots, \theta_n\right\} \subset [0, 2\pi]$,…

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In…

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our…

The structure of maximal faces of the cone of completely positive matrices is still not well understood in higher dimensions, mainly due to the lack of a general characterization of extreme exposed rays of the copositive cone beyond small…

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of…

Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of $G^\sigma$ is the same…

We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $\beta(A)=\frac{1}{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty$. Steinerberger [1]…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

Experimental data have provided stringent constraints on neutrino mixing parameters. In the standard parameterization the mixing angle $\theta_{23}$ is close to $\pi/4$. There are also evidences show that the CP violating phase is close to…

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the…

We investigate a relation among neutrino observables, three mixing angles and two mass squared differences, from a cascade texture of neutrino mass matrix. We show an allowed region of the correlation by use of current data of neutrino…

P. Erd\H{o}s, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$, which have order $n$ and minimum degree $\delta$.…

We propose a new mixing pattern for neutrinos with a nonzero mixing angle $\theta_{13}$. Under a simple form, it agrees well with current neutrino oscillation data and displays a number of intriguing features including the $\mu$-$\tau$…

Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets $R$ and $B$ with $|R|=|B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total \emph{squared} Euclidean distance of the…

Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erd\"{o}s-R\'{e}nyi model to establish \emph{critical thresholds}. Critical thresholds define narrow…

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…

For a neutrino mass matrix whose texture has an approximate flavor symmetry and where one has near degenerate neutrino mass, it is shown that the tribimaximal values for atmospheric angle $\sin^{2}{\theta_{23}}=1/2$ and solar angle…

The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta <1$. In…