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Keywords:
Laplacian eigenvalues; linear spread; ratio spread
Summary:
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.
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