A Note on the Signless Laplacian and Distance Signless Laplacian Eigenvalues of Graphs
Received:February 17, 2014  Revised:June 30, 2014
Key Words: signless Laplacian   distance signless Laplacian   spectral radius   eigenvalues.  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11171343).
Author NameAffiliation
Fenglei TIAN Department of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China 
Xiaoming LI Department of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China 
Jianling ROU Department of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China 
Hits: 3576
Download times: 3303
Abstract:
      Let $G$ be a simple graph. We first show that $ \delta_i\geq d_i-\sqrt{\lfloor \frac{i}{2} \rfloor \lceil \frac{i}{2} \rceil}$, where $\delta_i$ and $d_i$ denote the $i$-th signless Laplacian eigenvalue and the $i$-th degree of vertex in $G$, respectively. Suppose $G$ is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from $G$. In addition, for the distance signless Laplacian spectral radius $\rho_{\mathcal{Q}}(G)$, we determine the extremal graphs with the minimum $\rho_{\mathcal{Q}}(G)$ among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.
Citation:
DOI:10.3770/j.issn:2095-2651.2014.06.003
View Full Text  View/Add Comment