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arXiv · 2412.19203

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

Kun Cheng·Shuchao Li
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Abstract

Let $λ_2$ be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2023, Li and Sun \cite{LiSun1} determined all the connected $\{K_{2,3}, K_4\}$-minor free graphs whose second largest eigenvalue $λ_2\le 1$. As a continuance of it, in this paper we completely identify all the connected $\{K_5,K_{3,3}\}$-minor free graphs without $C_3$ whose second largest eigenvalue does not exceed 1. This partially solves an open problem posed by Li and Sun \cite{LiSun1}: 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.

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Kun Cheng, Shuchao Li. 2024-12-26. Connected triangle-free planar graphs whose second largest eigenvalue is at most 1. https://doi.org/10.1007/s40314-024-03027-4

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