Edge and spectral conditions for rainbow pancyclicity in graph collections

Let $\mathbf{G}=\{G_1,\dots,G_{n}\}$ be a collection of not necessarily distinct $n$-vertex graphs with a common vertex set $V$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq n$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(C)\to [n]$ such that $e\in E(G_{ϕ(e