Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical ideals of signed graphs with twin vertices   This paper studies critical ideals of graphs with twin vertices, which are
vertices with the same neighbors. A pair of such vertices are called replicated
if they are adjacent, and duplicated, otherwise. Critical ideals of graphs
having twin vertices have good properties and show regular patterns. Given a
graph $G=(V,E)$ and ${\bf d}\in \mathbb{Z}^{|V|}$, let $G^{\bf d}$ be the graph
obtained from $G$ by duplicating ${\bf d}_v$ times or replicating $-{\bf d}_v$
times the vertex $v$ when ${\bf d}_v>0$ or ${\bf d}_v<0$, respectively.
Moreover, given $\delta\in \{0,1,-1\}^{|V|}$, let \[
\mathcal{T}_{\delta}(G)=\{G^{\bf d}: {\bf d}\in \mathbb{Z}^{|V|} \text{ such
that } {\bf d}_v=0 \text{ if and only if }\delta_v=0 \text{ and } {\bf
d}_v\delta_v>0 \text{ otherwise}\} \] be the set of graphs sharing the same
pattern of duplication or replication of vertices. More than one half of the
critical ideals of a graph in $\mathcal{T}_{\delta}(G)$ can be determined by
the critical ideals of $G$. The algebraic co-rank of a graph $G$ is the maximum
integer $i$ such that the $i$-{\it th} critical ideal of $G$ is trivial. We
show that the algebraic co-rank of any graph in $\mathcal{T}_{\delta}(G)$ is
equal to the algebraic co-rank of $G^{\delta}$. For a large enough ${\bf d}\in
\mathbb{Z}^{V(G)}$, we show that the critical ideals of $G^{\bf d}$ have
similar behavior to the critical ideals of the disjoint union of $G$ and some
set $\{K_{n_v}\}_{\{v\in V(G)| d_v<0\}}$ of complete graphs and some set
$\{T_{n_v}\}_{\{v\in V(G) \, |\, {\bf d}_v>0\}}$ of trivial graphs.
Additionally, we pose important conjectures on the distribution of the
algebraic co-rank of the graphs with twins vertices. These conjectures imply
that twin-free graphs have a large algebraic co-rank, meanwhile a graph having
small algebraic co-rank has at least one pair of twin vertices.

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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: physics