Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Comparing Graphs of Different Sizes   We consider two notions describing how one finite graph may be larger than
another. Using them, we prove several theorems for such pairs that compare the
number of spanning trees, the return probabilities of random walks, and the
number of independent sets, among other combinatorial quantities. Our methods
involve inequalities for determinants, for traces of functions of operators,
and for entropy.

4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess