The Laplacian eigenvalue 2 of bicyclic graphs

Doost Ali Mojdeh; Mohammad Habibi; Masoumeh Farkhondeh

paper

The Laplacian eigenvalue 2 of bicyclic graphs

arXiv:1909.06578

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