Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral characterization of mixed extensions of small graphs   A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by
replacing each vertex of $G$ by a clique or a coclique, where vertices of $H$
coming from different vertices of $G$ are adjacent if and only if the original
vertices are adjacent in $G$. If $G$ has no more than three vertices, $H$ has
all but at most three adjacency eigenvalues equal to $0$ or $-1$. In this paper
we consider the converse problem, and determine the class $\cal G$ of all
graphs with at most three eigenvalues unequal to $0$ and $-1$. Ignoring
isolated vertices, we find that $\cal G$ consists of all mixed extensions of
graphs on at most three vertices together with some particular mixed extensions
of the paths $P_4$ and $P_5$.

18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph