Characterization of Outerplanar Graphs Whose Second Largest Eigenvalue is at Most 1

摘要

Let lambda(2) be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2021, Liu, Chen and Stanic determined all the connected {K-1,K-3, K-5-e}-free graphs whose second largest eigenvalue lambda(2) <= 1. In this paper, we completely identify all the connected {K-2,K-3, K-4}-minor free graphs whose second largest eigenvalue does not exceed 1. That is, we characterize all the connected outerplanar graphs satisfying lambda(2) <= 1. Furthermore, all the maximal outerplanar graphs having the same property can be deduced by our result obtained in this paper. Our main tools include analyzing the local structure of the outerplanar graph with respect to its girth.

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