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Open AccessArticle

Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

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Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran
2
Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran
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Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1233; https://doi.org/10.3390/math7121233
Received: 13 November 2019 / Revised: 8 December 2019 / Accepted: 9 December 2019 / Published: 12 December 2019
If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graph G = G 1 u v G 2 with V ( G ) = V ( G 1 ) V ( G 2 ) and E ( G ) = E ( G 1 ) E ( G 2 ) { e = u v } where u V ( G 1 ) and v V ( G 2 ) . In this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 u v G 2 . We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2. View Full-Text
Keywords: laplacian eigenvalue; multiplicity; eigenvector; unicyclic graph; bicyclic graph laplacian eigenvalue; multiplicity; eigenvector; unicyclic graph; bicyclic graph
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Farkhondeh, M.; Habibi, M.; Mojdeh, D.A.; Rao, Y. Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues. Mathematics 2019, 7, 1233.

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