February 9, 2023

# Doost Ali Mojdeh

Degree: Professor
Address: Department of Mathematics, University of Mazandaran, Babolsar, Iran
Education: Ph.D in Mathematics (Graph Theory and Combinatorics)
Phone: 011-35302448
Faculty: Faculty of Mathematical Sciences

## Research

 Title Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues Type Article Keywords Laplacian eigenvalue, multiplicity, eigenvector, unicyclic grap, bicyclic graph Journal Mathematics DOI doi:10.3390/math7121233 Researchers a { color: #4f98b0; } a:hover { color: #ffab00; } a:link:visited { text-decoration: none; } Masoumeh Farkhondeh (First researcher) , Mohammad Habibi (Second researcher) , Doost Ali Mojdeh (Third researcher) , Yongsheng Rao (Fourth researcher)

## Abstract

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 $2$.