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 [\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 $1$.
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