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A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré-Birkhoff-Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime.

A signed graph $\Gamma$ is a graph whose edges are labeled by signs. If $\Gamma$ has $n$ vertices, its spectral radius is the number $\rho \left(\Gamma \right):=max\left\{\right|{\lambda }_i\left(\Gamma \right)|:1\le i\le n\}$, where ${\lambda }_1\left(\Gamma \right)\ge \cdots \ge {\lambda }_n\left(\Gamma \right)$ are the eigenvalues of the signed adjacency matrix $A\left(\Gamma \right)$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes ${𝔘}_n$ and ${𝔅}_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.