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Keywords:
Laplacian graph spectra; bipartite graph; spread of graph
Laplacian graph spectra; bipartite graph; spread of graph
Summary:
Let $G=(V,E)$, $V=\{v_1,v_2,\ldots ,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\geq d_2\geq \cdots \geq d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^+=D+A$ and the normalized signless Laplacian matrix as $\mathcal {L}^+=D^{-1/2}L^+ D^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r(G)= \gamma _{2}^{+}/ \gamma _{n}^{+}$ and $l(G)=\gamma _{2}^{+}-\gamma _{n}^{+}$, where $\gamma _1^+ \ge \gamma _2^+\ge \cdots \ge \gamma _n^+ \ge 0$ are eigenvalues of $\mathcal {L}^+$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads of connected graphs. In addition, we present a better lower bound on the signless Laplacian spread.
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