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 graphIn this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 uv 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.

If G is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graphIn this paper, we consider the eigenvector of unicycle graphs. We study the relationship between the Laplacian eigenvalue 2 of unicyclic graphs G 1 and G 2 ; and bicyclic graphs G = G 1 ⊙ G 2 . We also characterize the broken sun graphs and the one edge connection of two broken sun graphs by their Laplacian eigenvalue 2.