3 problems

Let $t>r\geq2$ and suppose $(r-1)^r\mid n-r$. Let $H^*$ be a linear $r$-graph attaining the spectral extremal value for Berge-$K_{3,t}$-free linear $r$-graphs, and let…

Let $G=(V,E)$ be an ordinary graph, let $k=2r$ be even, and let $G^k=(V^k,E^k)$ be the $k$-power hypergraph of $G$. Let $\mathcal L^k$ and $\mathcal Q^k$ be the Laplacian and signl…

Hyperstar characterization conjecture. The largest Laplacian H-eigenvalue $\lambda(\mathcal L)$ is equal to $d$ if and only if $G$ is a hyperstar.