Planar graphs whose second largest eigenvalue does not exceed 1

The problem posed by Hoffman, namely, to characterize graphs whose second largest adjacency eigenvalue is at most $1$, was first studied by D.~M.~Cvetković in 1982 [$\mathit{\text{Publ.}}\mathit{\text{Inst.}}\mathit{\text{Math.}}\mathit{\text{(Beograd)}}$ $\mathbf{\text{31}}$ (1982), 15–20]. In this paper, we completely characterize all connected planar graphs whose second largest eigenvalue does not exceed $1$.