Abstract
Let \(\lambda _2\) be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2021, Liu, Chen and Stanić determined all the connected \(\{K_{1,3}, K_5 -e\}\)-free graphs whose second largest eigenvalue \(\lambda _2\leqslant 1\). In this paper, we completely identify all the connected \(\{K_{2,3},K_4\}\)-minor free graphs whose second largest eigenvalue does not exceed 1. That is, we characterize all the connected outerplanar graphs satisfying \(\lambda _2\leqslant 1\). 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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Acknowledgements
We are very grateful to all the referees for their many helpful comments. In particular, for the suggestion of using the result in [21] which substantially shortened the presentation and for the cleaner and shorter proofs of some technical statements. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164) and the excellent doctoral dissertation cultivation grant from Central China Normal University (Grant No. 2022YBZZ033).
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Li, S., Sun, W. Characterization of Outerplanar Graphs Whose Second Largest Eigenvalue is at Most 1. Results Math 78, 104 (2023). https://doi.org/10.1007/s00025-023-01883-w
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DOI: https://doi.org/10.1007/s00025-023-01883-w