Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Max Min vertex cover and the size of Betti tables   Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated
vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In
this paper, we study the pair of integers that measure the projective dimension
and the regularity of $S/I(G)$. We show that if the projective dimension of
$S/I(G)$ attains its minimum value $2\sqrt{n}-2$ then, with only one exception,
the its regularity must be 1. We also provide a full description for the
spectrum of the projective dimension of $S/I(G)$ when the regularity attains
its minimum value 1.

35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas