Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Achievable multiplicity partitions in the inverse eigenvalue problem of
  a graph   Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued
symmetric matrices whose off-diagonal entries are nonzero precisely when the
corresponding vertices of the graph are adjacent, and the diagonal entries are
free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the
eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a
multiplicity partition.
  We study graphs for which a multiplicity partition with only two integers is
possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with
partitions $[n-2,2]$ have been characterized. We find families of graphs $G$
for which there is a matrix in $\mathcal{S}(G)$ with multiplicity partition
$[n-k,k]$ for $k\geq 2$. We focus on generalizations of the complete
multipartite graphs. We provide some methods to construct families of graphs
with given multiplicity partitions starting from smaller such graphs. We also
give constructions for graphs with matrix in $\mathcal{S}(G)$ with multiplicity
partition $[n-k,k]$ to show the complexities of characterizing these graphs.

1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int