Ordering quasi-tree graphs by the second largest signless Laplacian eigenvalues
Received:September 04, 2019  Revised:February 19, 2020
Key Word: Quasi-tree graph   Signless Laplacian matrix   Second largest eigenvalue   Sum of eigenvalues   Ordering
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 Author Name Affiliation E-mail Zhen Lin China University of Mining and Technology LNLinZhen@163.com Shu-Guang Guo Yancheng Teachers University ychgsg@163.com Lianying Miao China University of Mining and Technology miaolianying@cumt.edu.cn
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A connected graph $G=(V, E)$ is called a quasi-tree graph if there exists a vertex $v_0\in V(G)$ such that $G-v_0$ is a tree. In this paper, we determine all quasi-tree graphs of order $n$ with the second largest signless Laplacian eigenvalue greater than or equal to $n-3$. As an application, we determine all quasi-tree graphs of order $n$ with the sum of the two largest signless Laplacian eigenvalues greater than to $2n-\frac{5}{4}$.