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 $G=G_{1}\odot_{uv} G_{2}$ with $V(G)=V(G_{1})\cup V(G_{2})$ and $E(G)= E(G_{1})\cup E(G_{2})\cup \{e=uv\}$ where $u\in V(G_1)$ and $v\in V(G_2)$. In this paper, we study some structural conditions ensuring the presence of $2$ in the Laplacian spectrum of bicyclic graphs of type $G_1\odot_{uv} G_2$. 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 $2$.
