Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some New Results Concerning Power Graphs and Enhanced Power Graphs of
  Groups   The directed power graph $\vec{\mathcal G}(\mathbf G)$ of a group $\mathbf G$
is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is
a power of $x$. The power graph of $\mathbf G$, denoted by $\mathcal G(\mathbf
G)$, is the underlying simple graph. The enhanced power graph $\mathcal
G_e(\mathbf G)$ of $\mathbf G$ is the simple graph with vertex set $G$ in which
two elements are adjacent if they generate a cyclic subgroup. In this paper it
is proved that, if two groups have isomorphic power graphs, then they have
isomorphic enhanced power graphs too. A sufficient condition for a group to
have perfect enhanced power graph is given. By this condition, any group having
order divisible by at most two primes has perfect enhanced power graph.

26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph