Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimal Sum Labeling of Graphs   A graph $G$ is called a sum graph if there is a so-called sum labeling of
$G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that
for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists
a vertex $w\in V(G)$ such that $\ell(u)+\ell(v) = \ell(w)$. We say that sum
labeling $\ell$ is minimal if there is a vertex $u\in V(G)$ such that
$\ell(u)=1$. In this paper, we show that if we relax the conditions (either
allow non-injective labelings or consider graphs with loops) then there are sum
graphs without a minimal labeling, which partially answers the question posed
by Miller, Ryan and Smyth in 1998.

22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int