Planar graphs whose second largest eigenvalue does not exceed 1

Source abstract

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 [\textit{Publ. Inst. Math. (Beograd)} \textbf{31} (1982), 15--20]. In this paper, we completely characterize all connected planar graphs whose second largest eigenvalue does not exceed $1$.