Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On singular signed graphs with nullspace spanned by a full vector:
  Signed nut graphs   A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed
sign-balanced if it is equivalent to an unsigned graph under the operation of
sign switching; otherwise it is called sign-unbalanced. A nut graph has a one
dimensional kernel with a corresponding eigenvector that is full. In this paper
we generalise the notion of nut graphs to signed graphs. Orders for which
unsigned regular nut graphs exist were determined recently for the degrees up
to $11$. By extending the definition to signed nut graphs, we find all pairs
$(\rho, n)$ for which a $\rho$-regular nut graph (sign-balanced or
sign-unbalanced) of order $n$ exists with $\rho \le 11$. We devise a
construction for signed nut graphs based on a smaller `seed' graph, giving
infinite series of both sign-balanced and sign-unbalanced $\rho$-regular nut
graphs. All orders for which a complete sign-unbalanced nut graph exists are
characterised; they have underlying graph $K_n$ with $n \equiv 1 \pmod 4$. All
orders for which a regular sign-unbalanced nut graph with $\rho = n - 2$ exists
are also characterised; they have an underlying cocktail-party graph
$\mathrm{CP}(n)$ with even $n \geq 8$.

18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph