Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Wiener Index of Signed Graphs   The Wiener index of a graph $W(G)$ is a well studied topological index for
graphs. An outstanding problem of \v{S}olt{\'e}s is to find graphs $G$ such
that $W(G)=W(G-v)$ for all vertices $v\in V(G)$, with the only known example
being $G=C_{11}$. We relax this problem by defining a notion of Wiener indices
for signed graphs, which we denote by $W_\sigma(G)$, and under this relaxation
we construct many signed graphs such that $W_\sigma(G)=W_\sigma(G-v)$ for all
$v\in V(G)$. This ends up being related to a problem of independent interest,
which asks when it is possible to $2$-color the edges of a graph $G$ such that
there is a path between any two vertices of $G$ which uses each color the same
number of times.

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess