Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues.

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. [ABSTRACT FROM AUTHOR]