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.
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Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11171343). |
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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 |
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