Ordering QuasiTree Graphs by the Second Largest Signless Laplacian Eigenvalues 
Received:September 04, 2019 Revised:March 17, 2020 
Key Words:
quasitree graph signless Laplacian matrix second largest eigenvalue sum of eigenvalues ordering

Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11771443), the Fundamental Research Funds for the Central Universities (Grant No.2018BSCXB24) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No.KYCX18_1980). 

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Abstract: 
A connected graph $G=(V, E)$ is called a quasitree graph if there exists a vertex $v_0\in V(G)$ such that $Gv_0$ is a tree. In this paper, we determine all quasitree graphs of order $n$ with the second largest signless Laplacian eigenvalue greater than or equal to $n3$. As an application, we determine all quasitree graphs of order $n$ with the sum of the two largest signless Laplacian eigenvalues greater than to $2n\frac{5}{4}$. 
Citation: 
DOI:10.3770/j.issn:20952651.2020.05.002 
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