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Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

Author

Listed:
  • 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)

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.

Suggested Citation

  • Masoumeh Farkhondeh & Mohammad Habibi & Doost Ali Mojdeh & Yongsheng Rao, 2019. "Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues," Mathematics, MDPI, vol. 7(12), pages 1-9, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1233-:d:297198
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