Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A lower bound for the sum of the two largest signless Laplacian
  eigenvalues   Let $G$ be a graph of order $n \geq 3$ with sequence degree given as
$d_{1}(G) \geq ... \geq d_{n}(G)$ and let $\mu_1(G),..., \mu_n(G)$ and $q_1(G),
..., q_{n}(G)$ be the Laplacian and signless Laplacian eigenvalues of $G$
arranged in non increasing order, respectively. Here, we consider the Grone's
inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin.
Multilin. Alg. 39 (1995) 133--136] $$ \sum_{i=1}^{k} \mu_{i}(G) \geq
\sum_{i=1}^{k} d_{i}(G)+1$$ and prove that for $k=2$, the equality holds if and
only if $G$ is the star graph $S_{n}.$ The signless Laplacian version of
Grone's inequality is known to be true when $k=1.$ In this paper, we prove that
it is also true for $k=2,$ that is, $$q_{1}(G)+q_{2}(G) \geq d_1(G)+d_2(G)+1$$
with equality if and only if $G$ is the star $S_{n}$ or the complete graph
$K_{3}.$ When $k \geq 3$, we show a counterexample.

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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th