Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiset Dimensions of Trees   Let $G$ be a connected graph and $W$ be a set of vertices of $G$. The
representation multiset of a vertex $v$ with respect to $W$, $r_m (v|W)$, is
defined as a multiset of distances between $v$ and the vertices in $W$. If $r_m
(u |W) \neq r_m(v|W)$ for every pair of distinct vertices $u$ and $v$, then $W$
is called an m-resolving set of $G$. If $G$ has an m-resolving set, then the
cardinality of a smallest m-resolving set is called the multiset dimension of
$G$, denoted by $md(G)$; otherwise, we say that $md(G) = \infty$.
  In this paper, we show that for a tree $T$ of diameter at least 2, if $md(T)
< \infty$, then $md(T) \leq n-2$. We conjecture that this bound is not sharp in
general and propose a sharp upper bound. We shall also provide necessary and
sufficient conditions for caterpillars and lobsters having finite multiset
dimension. Our results partially settled a conjecture and an open problem
proposed in [4].

11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin