Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Co-Maximal Subgroup Graph of a Group   The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose
vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$
are adjacent if $HK=G$. In this paper, we continue the study of $\Gamma(G)$,
especially when $\Gamma(G)$ has isolated vertices. We define a new graph
$\Gamma^*(G)$, which is obtained by removing isolated vertices from
$\Gamma(G)$. We characterize when $\Gamma^*(G)$ is connected, a complete graph,
star graph, has an universal vertex etc. We also find various graph parameters
like diameter, girth, bipartiteness etc. in terms of properties of $G$.

12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess