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

TL;DR

This paper characterizes connected planar graphs with no triangles and specific minors whose second largest eigenvalue is at most 1, extending previous classifications and partially solving an open problem.

Contribution

It completely identifies certain connected planar graphs without triangles and specific minors with second eigenvalue ≤ 1, advancing spectral graph theory.

Findings

Classified all such graphs with second eigenvalue ≤ 1

Extended previous work on minor-free graphs

Partially solved an open problem in spectral graph theory

Abstract

Let $\lambda_2$ 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 $\lambda_2\le 1$. 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.