The Laplacian eigenvalue 2 of bicyclic graphs

Abstract

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 graph $G=G_{1}\odot G_{2}$ 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 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 $G=G_{1}\odot G_{2}$. We also characterize the broken sun graphs and the one edge connection of two broken sun graphs by their Laplacian eigenvalue $2$.