Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the metric subgraphs of a graph   The three subgraphs of a connected graph induced by the center, annulus and
periphery are called its metric subgraphs. The main results are as follows. (1)
There exists a graph of order $n$ whose metric subgraphs are all paths if and
only if $n\ge 13$ and the smallest size of such a graph of order $13$ is $22;$
(2) there exists a graph of order $n$ whose metric subgraphs are all cycles if
and only if $n\ge 15,$ and there are exactly three such graphs of order $15;$
(3) for every integer $k\ge 3,$ we determine the possible orders for the
existence of a graph whose metric subgraphs are all connected $k$-regular
graphs; (4) there exists a graph of order $n$ whose metric subgraphs are
connected and pairwise isomorphic if and only if $n\ge 24$ and $n$ is divisible
by $3.$ An unsolved problem is posed.

29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs