Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

Masoumeh Farkhondeh, Mohammad Habibi, Doost Ali Mojdeh and Yongsheng Rao
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Masoumeh Farkhondeh: Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran
Mohammad Habibi: Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran
Doost Ali Mojdeh: Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran
Yongsheng Rao: Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China

Mathematics, 2019, vol. 7, issue 12, 1-9

Abstract: 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.

Keywords: laplacian eigenvalue; multiplicity; eigenvector; unicyclic graph; bicyclic graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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