Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Perfect matching and distance spectral radius in graphs and bipartite
  graphs   A perfect matching in a graph $G$ is a set of nonadjacent edges covering
every vertex of $G$. Motivated by recent progress on the relations between the
eigenvalues and the matching number of a graph, in this paper, we aim to
present a distance spectral radius condition to guarantee the existence of a
perfect matching. Let $G$ be an $n$-vertex connected graph where $n$ is even
and $\lambda_{1}(D(G))$ be the distance spectral radius of $G$. Then the
following statements are true.
  \noindent$\rm{I)}$ If $4\le n\le10$ and ${\lambda }_{1} (D\left(G\right))\le
{\lambda }_{1} (D(S_{n,{\frac{n}{2}}-1}))$, then $G$ contains a perfect
matching unless $G\cong S_{n,{\frac{n}{2}-1}}$ where
$S_{n,{\frac{n}{2}-1}}\cong K_{{\frac{n}{2}-1}}\vee ({\frac{n}{2}+1})K_1$.
  \noindent$\rm{II)}$ If $n\ge 12$ and ${\lambda }_{1} (D\left(G\right))\le
{\lambda }_{1} (D(G^*))$, then $G$ contains a perfect matching unless $G\cong
G^*$ where $G^*\cong K_1\vee (K_{n-3}\cup2K_1)$.
  Moreover, if $G$ is a connected $2n$-vertex balanced bipartite graph with
$\lambda_{1}(D(G))\le \lambda_{1}(D(B_{n-1,n-2})) $, then $G$ contains a
perfect matching, unless $G\cong B_{n-1,n-2}$ where $B_{n-1,n-2}$ is obtained
from $K_{n,n-2}$ by attaching two pendent vertices to a vertex in the
$n$-vertex part.

25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat