Connected triangle-free planar graphs whose second largest eigenvalue is at most 1

Let \(\lambda _2\) ${\lambda }_2$ be the second largest eigenvalue of the adjacency matrix of a connected graph. Li and Sun (Results Math 78:104, 2023) determined all the connected \(\{K_{2,3}, K_4\}\) $\{K_{2,3},K_4\}$ -minor free graphs whose second largest eigenvalue \(\lambda _2\le 1\) ${\lambda }_2\le 1$ . As a continuance of it, in this paper we completely identify all the connected \(\{K_5,K_{3,3}\}\) $\{K_5,K_{3,3}\}$ -minor free graphs without \(C_3\) $C_3$ whose second largest eigenvalue does not exceed 1. This partially solves an open problem posed by Li and Sun (2023): characterize all connected planar graphs whose second largest eigenvalue is at most 1. Our main tools include the spectral theory and the local structure characterization of the planar graph with respect to its girth.