Rainbow Hamiltonicity and the spectral radius

Let $G=\{G_1,\dots ,G_n\}$ be a family of graphs of order n with the same vertex set. A rainbow Hamiltonian cycle in $G$ is a cycle that visits each vertex precisely once such that any two edges belong to different graphs of $G$. We show that if each $G_i$ has more than $\left(\begin{array}{c}n-1\\ 2\end{array}\right)+1$ edges, then $G$ admits a rainbow Hamiltonian cycle and pose the problem of characterizing rainbow Hamiltonicity under the condition that all $G_i$ have at least $\left(\begin{array}{c}n-1\\ 2\end{array}\right)+1$ edges. Towards a solution of that problem, we give a sufficient condition for the existence of a rainbow Hamiltonian cycle in terms of the spectral radii of the graphs in $G$ and completely characterize the corresponding extremal graphs.