Let $\mathcal{G}_{n,\beta}$ be the set of simple graphs of order $n$ with given matching number $\beta$. In this paper, we investigate the maximal signless Laplacian spectral radius in $\mathcal{G}_{n,\beta}$ and characterize the extremal graphs with maximal signless Laplacian spectral radius.

Under some conditions on a Hilbert space $H$ of analytic functions on the open unit disc we will show that for every nontrivial invariant subspace $\mathcal{M}$ of $H$, there exists a unique nonconstant inner function $\varphi$ such that $\mathcal{M}=\varphi H$. This extends the Beurling’s Theorem.

Let $f:X\to Y$ be a covering of closed oriented surfaces. Then $f$ induces a homomorphism $f_{*} :H_{1} (X,\mathbf{Z})\to H_{1} (Y,\mathbf{Z})$ of the first homology groups. We consider the converse and characterize—in terms of matrices—the abstractly given homomorphisms of the first homology groups which can be induced by coverings of prime degree. We also classify the induced homomorphisms in these cases.