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Two inequalities for the Laplacian spread of graphs are proved in this note. These inequalities are reverse to those obtained by Z. You, B. Liu: The Laplacian spread of graphs, Czech. Math. J. 62 (2012), 155–168.

Let $G=(V,E)$, $V=\{v_1,v_2,...,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. 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 ${ℒ}^{+}=D^{-1/2}{L}^{+}{D}^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r\left(G\right)={\gamma }_2^{+}/{\gamma }_n^{+}$ and $l\left(G\right)={\gamma }_2^{+}-{\gamma }_n^{+}$, where ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$ are eigenvalues of ${ℒ}^{+}$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads...

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Let $G$ be an undirected connected graph with $n$, $n\ge 3$, vertices and $m$ edges with Laplacian eigenvalues ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{n-1}>{\mu }_n=0$. Denote by ${\mu }_I={\mu }_{r_1}+{\mu }_{r_2}+\cdots +{\mu }_{r_k}$, $1\le k\le n-2$, $1\le r_1<r_2<\cdots <r_k\le n-1$, the sum of $k$ arbitrary Laplacian eigenvalues, with ${\mu }_{I_1}={\mu }_1+{\mu }_2+\cdots +{\mu }_k$ and ${\mu }_{I_n}={\mu }_{n-k}+\cdots +{\mu }_{n-1}$. Lower bounds of graph invariants ${\mu }_{I_1}-{\mu }_{I_n}$ and ${\mu }_{I_1}/{\mu }_{I_n}$ are obtained. Some known inequalities follow as a special case.