For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes path-length distance. Describing set $\pi(G)$ consisting those $q\in \mathbb{R}$ for which $Q_q$ positive semidefinite fundamental in asymptotic spectral analysis graphs from viewpoint quantum probability theory. Assume that $G$ has at least two ve...

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call matrix in this $k$-locally In order compare PSD matrices, study eigenvalues matrices. T...

Given a matrix family, it is interesting to know whether some important properties or structures of the class of matrices are inherited by their submatrices or by the matrices associated with the original matrices. It is known that the principal submatrices and the Schur complements of positive semidefinite matrices are positive semidefinite matrices; the same is true of M-matrices, H-matrices,...