5 problems

An $\ell$-lift of $K_n$ is obtained by replacing every vertex of $K_n$ with an independent set of size $\ell$ and every edge with a matching of size $\ell$ between the correspondin…

Let $H$ be a $d$-regular multigraph on $k$ vertices, and let $\varepsilon>0$. Write $\mathcal{L}_{m}(H)$ for an $m$-sheeted lift of $H$. The operators…

Let $G$ be a finite regular graph, and let $H$ be a 2-lift of $G$. An eigenvalue of $H$ not inherited from $G$ is called new, and let $\rho(\mathsf{UCT}(G))$ denote the relevant un…

Let $G$ be a finite graph, and let $H$ be a lift of $G$. Write $\mathsf{spec}(G)$ for the spectrum of $G$, and let $\rho(\mathsf{UCT}(G))$ denote the relevant universal-cover spect…

Ramanujan-lift probability conjecture. There exists a constant $c$ with $0<c<1$ such that