Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pairs of a tree and a nontree graph with the same status sequence   The status of a vertex $x$ in a graph is the sum of the distances between $x$
and all other vertices. Let $G$ be a connected graph. The status sequence of
$G$ is the list of the statuses of all vertices arranged in nondecreasing
order. $G$ is called status injective if all the statuses of its vertices are
distinct. Let $G$ be a member of a family of graphs $\mathscr{F}$ and let the
status sequence of $G$ be $s.$ $G$ is said to be status unique in $\mathscr{F}$
if $G$ is the unique graph in $\mathscr{F}$ whose status sequence is $s.$ In
2011, J.L. Shang and C. Lin posed the following two conjectures. Conjecture 1:
A tree and a nontree graph cannot have the same status sequence. Conjecture 2:
Any status injective tree is status unique in all connected graphs. We settle
these two conjectures negatively. For every integer $n\ge 10,$ we construct a
tree $T_n$ and a unicyclic graph $U_n,$ both of order $n,$ with the following
two properties: (1) $T_n$ and $U_n$ have the same status sequence; (2) for
$n\ge 15,$ if $n$ is congruent to $3$ modulo $4$ then $T_n$ is status injective
and among any four consecutive even orders, there is at least one order $n$
such that $T_n$ is status injective.

25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con